IN  MEMORIAM 
FLORIAN  CAJORI 


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COLLEGE   ALGEBRA 


BY 


J.    M.   TAYLOR.   A.M.,    LL.D., 
n 

PROFESSOR  OF   MATHEMATICS    IN    COLGATE    UNIVERSITY 


SIXTH  EDITION 


ALLYN    AND    BACON 

Boston  antj  Cf)icago 


Copyright,  1889, 
By  Allyn  and  Bacon. 


Berwick  &  Smith,  Norwood,  Mass.,  U.i 


PREFACE 


THIS  work  originated  in  the  author's  desire  for  a 
course  in  Algebra  suited  to  the  needs  of  his 
own  pupils.  The  increasing  claims  of  new  sciences 
to  a  place  in  the  college  curriculum  render  necessary 
a  careful  selection  of  matter  and  the  most  direct 
methods  in  the  old.  The  author's  aim  has  been  to 
present  each  subject  as  concisely  as  a  clear  and 
rigorous  treatment  would  allow. 

The  First  Part  embraces  an  outline  of  those  fun- 
damental principles  of  the  science  that  are  usually 
required  for  admission  to  a  college  or  scientific 
school.  The  subjects  of  Equivalent  Equations  and 
Equivalent  Systems  of  Equations  are  presented  more 
fully  than  others.  Until  these  subjects  are  more 
scientifically  understood  by  the  average  student,  it 
will  be  found  profitable  to  review  at  least  this  por- 
tion of  the  First  Part. 

In  the  Second  Part  a  full  discussion  of  the  Theory 
of  Limits  followed  by  one  of  its  most  important  ap- 
plications, Differentiation,  leads  to  clear  and  concise 


IV  PREFACE. 

proofs  of  the  Binomial  Theorem,  Logarithmic  Series, 
and  Exponential  Series,  as  particular  cases  of  Mac- 
laurin's  Formula.  It  also  affords  the  student  an  easy 
introduction  to  the  concepts  and  methods  of  the 
higher  mathematics. 

Each  chapter  is  as  nearly  as  possible  complete  in 
itself,  so  that  the  order  of  their  succession  can  be 
varied  at  the  discretion  of  the  teacher;  and  it  is 
recommended  that  Summation  of  Series,  Continued 
Fractions,  and  the  sections  marked  by  an  asterisk 
be  reserved  for  a  second  reading. 

In  writing  these  pages  the  author  has  consulted 
especially  the  works  of  Laurent,  Bertrand,  Serret, 
Chrystal,  Hall  and  Knight,  Todhunter,  and  Burnside 
and  Panton.  From  these  sources  many  of  the  prob- 
lems and  examples  have  been  obtained. 

J.  M.  TAYLOR. 
Hamilton,  N.  Y.,  1889. 


PREFACE   TO   THIRD    EDITION. 

In  this  edition  a  number  of  changes  have  been 
made  in  both  definitions  and  demonstrations.  In 
the  Second  Part,  derivatives,  but  not  differentials, 
are  employed.  Two  chapters  have  been  added  ;  one 
on  Determinants,  the  other  on  the  Graphic  Solution 
of  Equations  and  of  Systems  of  Equations. 

j.  M.  TAYLOR. 

JiAMILTON,  N,  Y.,  1895. 


CONTENTS, 


FIRST   PART. 


CHARTER   I. 

Pagb 

Definitions  and  Notation 1-9 


CHARTER   n. 
Fundamental  Operations 10-21 

CHAPTER   HI. 
Fractions 22-24 

CHAPTER    IV. 
Theory  of  Exponents 25-32 

CHAPTER  V. 

Factoring .      33 

Highest  Common  Divisor 38 

Lowest  Common  Multiple 41 


VI  CONTENTS. 


CHAPTER  VI. 

Page 

Involution,  Evolution 42 

Surds  and  Imaginaries 46 


CHAPTER  VII. 

Equations S^ll 

Equivalent  Equations ^'j 

Linear  Equations 63 

Quadratic  and  Higher  Equations 65 

CHAPTER   VIII. 

Systems  of  Equations 78-91 

Equivalent  Systems 79 

Methods  of  Elimination 80 

Systems  of  Quadratic  Equations 85 

CHAPTER   IX. 

Indeterminate  Equations  and  Systems      ....      92 

Discussion  of  Problems 98 

Inequalities loi 

CHAPTER  X. 
Ratio,  Proportion,  and  Variation      ....    .104-114 

CHAPTER   XI. 
The  Progressions 115-121 


CONTENTS.  Vll 


SECOND    PART. 


CHAPTER  XII. 

Page 

Functions  and  Theory  of  Limits 122-133 

Functions  and  Functional  Notation 123 

Theory  of  Limits .125 

Vanishing  Fractions 132 

Incommensurable  Exponents 132 

CHAPTER  XIII. 

Derivatives I34-U7 

Derivatives I34 

Illustration  of  A  («:r2) 136 

Rules  for  finding  Derivatives      ........  137 

Successive  Derivatives MS 

Continuity 146 

CHAPTER   XIV. 

Development  of  Functions  in  Series     ....  148-170 

Development  by  Division I49 

Principles  of  Undetermined  Coefficients 150 

Resolution  of  Fractions  into  Partial  Fractions    ...  154 

Reversion  of  Series I59 

Maclaurin's  Formula 161 

The  Binomial  Theorem 163 


viii  CONTENTS. 


CHAPTER  XV. 

Page 
CONVERGENCY   AND    SUMMATION    OF    SERIES        .      .       .  I7I-I92 

Convergency  of  Series 171 

Recurring  Series 177 

Method  of  Differences 182 

Interpolation 188 


CHAPTER  XVI. 

Logarithms 193-214 

General  Principles 193 

Common  Logarithms 197 

Exponential  Equations 203 

The  Logarithmic  Series 205 

The  Exponential  Series 211 


CHAPTER  XVn. 
Compound  Interest  and  Annuities 215-223 


CHAPTER  XVHI. 
Permutations  and  Combinations 224-236 

CHAPTER  XIX. 

Probability , 237-253 

Single  Events 237 

Compoufid  Events 241 


CONTENTS.  IX 


CHAPTER   XX. 

Page 

Continued  Fractions 254-265 

Conversion  of  a  Fraction  into  a  Continued  Fraction  .     255 

Convergents 256 

Periodic  Continued  Fractions 263 


CHAPl^ER  XXI. 

Theory  of  Equations 266-317 

Reduction  to  the  Form  F(x)  =  0 266 

Divisibility  of /^(.r) 267 

Horner's  Method  of  Synthetic  Division 268 

Number  of  Roots 272 

Relations  between  Coeff  cients  and  Roots 274 

Imaginary  Roots 276 

Integral  Roots 280 

Limits  of  Roots 281 

Equal  Roots 284 

Change  of  Sign  of /'(r) 286 

Sturm's  Theorem 288 

Transformation  of  Equations 295 

Horner's  Method  of  Solving  Numerical  Equations     .  300 

Reciprocal  Equations 307 

Binomial  Equations 310 

Cubic  Equations 312 

Biquadratic  Equations 315 


CONTENTS. 


CHAPTER   XXII. 

Page 

Determinants 318-346 

Determinants  of  the  Second  Order 318 

Determinants  of  the  Third  Order 322 

Determinants  of  the  «th  Order 328 

Properties  of  Determinants 330 

Minors  and  Co-factors 336 

Expression  of  A  in  Co-factors 337 

Eliminants 340 

Multiplication  of  Determinants 343 


CHAPTER   XXIII. 

Graphic  Solution  of  Equations  and  of  Systems  347-363 

Co-ordinates  of  a  Point 348 

Graphic  Solution  of  Indeterminate  Equations   .     .     .  349 

Graphic  Solution  of  Systems  of  Equations  ....  353 
Properties  of  -^(:r)  and  of  /^(.r)  =  0  illustrated  by  the 

Graph  of /^  (;r) 357 

Geometric  Representat'on  of  Imaginary  and  Complex 

Numbers -  .  360 


ALGEBRA 


FIRST   PART. 


CHAPTER   I. 
DEFINITIONS  AND   NOTATION. 

1.  Quantity  is  anything  that  can  be  increased,  di- 
minished, or  measured;  as  any  portion  of  time  or 
space,  any  distance,  force,  or  weight. 

2.  To  measure  a  quantity  is  to  find  how  many 
times  it  contains  some  other  quantity  of  the  same 
kind  taken  as  a  unit,  or  standard  of  comparison. 

Thus,  to  measure  a  distance,  we  find  how  many  times  it  con- 
tains some  other  distance  taken  as  a  unit.  To  measure  a  por- 
tion of  time,  we  find  how  many  times  it  contains  some  other 
portion  of  time  taken  as  a  unit. 

3.  By  counting  the  units  in  a  quantity,  we  gain  the 
idea  of  '  how  many  ' ;  that  is,  of  Arithmetical  Number. 
If,  in  the  measure  of  any  quantity,  we  omit  the  unit 
of  measure,  we  obtain  an  aritJunctical  number.  It 
may  be  a  whole  number  or  a  fraction.  Thus  by 
omitting  the  units  ft.,  lb.,  hr.,  in  6  ft.,  3  lbs.,  and 
Yi  hr.,  we  obtain  the  whole  numbers  6  and  3,  and 
the  fraction  Yi, 


2  ALGEBRA. 

4.  Positive  and  Negative  Quantities.  Two  quanti- 
ties of  the  same  kind  are  opposite  in  qiuxlity,  if  when 
united,  any  amount  of  the  one  annuls  or  destroys  an 
equal  amount  of  the  other.  Of  two  opposites  one  is 
said  to  be  Positive  in  quality,  and  the  other  Negative. 

Thus,  credits  and  debits  are  opposites,  since  equal  amounts 
of  the  two  destroy  each  other.  If  we  call  credits  positive, 
debits  will  be  negative.  Two  forces  acting  along  the  same  line 
in  opposite  directions  are  opposites ;  if  we  call  one  positive,  the 
other  is  negative. 

5.  Algebraic  Number.  The  sign  +,  read  positive, 
and  — ,  read  i:eg  t've,  are  used  with  numbers,  or  their 
symbols,  to  denote  their  quality,  or  the  quality  of  the 
quantities  which  they  represent. 

Thus,  if  we  call  credit  positive,  +  $5  denotes  $5  of 
credit,  and  —  $4  denotes  $4  of  debt.  If  +  8  in.  de- 
notes 8  in.  to  the  right,  —9  in.  denotes  9  in.  to  the 
left.  By  omitting  the  particular  units  $  and  in.,  in 
+  $Sy  ~  ^4»  +  S  i"-»  ~  9  ^"•'  ^v^  obtain  the  algebraic 
numbers  +  5,  —  4,  +  8,  —  9.  +  5  is  read  '  positive  5,' 
—  4  is  read  '  negative  4.'  Each  of  these  numbers  has 
not  only  an  arithmetical  value^  but  also  the  quality  of 
one  of  two  opposites  ;  hence 

An  Algebraic  Number  is  one  that  has  both  an 
arithmetical  value  and  the  quality  of  one  of  two 
opposites. 

Two  algebraic  numbers  that  are  equal  in  arith- 
metical value  but  opposite  in  quality  destroy  each 
other  when  added. 


DEFINITIONS   AND   NOTATION.  3 

The  element  of  quality  in  algebraic  number  doubles 

the  range  of  number. 

Thus,  the  integers  of  arithmetic  make  up  the  simple 

scries, 

o,  I,  2,  3,  4,  5»  6,  7,  ...,  ^;  (i) 

while   the   integers  of  algebra  make   up   the  double 

series, 

-:»,..  .,-4,  -3,-2,  -I,  ±o,  +1,  +2,  +3,  +4,  •  •  •,+  ^-  (2) 

An  algebraic  number  is  said  to  be  i7tcreased  by 
adding  a  positive  number,  and  decreased  by  adding  a 
negative  number. 

If  in  series  (2)  we  add  +  i  to  any  number,  we  ob- 
tain the  next  right-hand  number.  Thus,  if  to  +  3  we 
add  -t-  I,  we  obtain  -I-4;  if  to  —4  we  add  4-  i>  we 
obtain  —  3 ;  and  so  on.  That  is,  positive  numbers 
increase  from  zero,  while  negative  numbers  decrease 
from  zero. 

Hence  positive  numbers  are  algebraically  the 
greater,  the  greater  their  arithmetical  values;  while 
negative  numbers  are  algebraically  the  less,  the 
greater  their  arithmetical  values. 

All  numbers  are  quantities,  and  the  term  quantity 
is  often  used  to  denote  number. 

6.  Symbols  of  Number.  Arithmetical  numbers  are 
usually  denoted  by  figures.  Algebraic  numbers 
are  denoted  by  letters,  or  by  figures  with  the  signs 
4:  and  —  prefixed  to  denote  their  quality.     A  letter 


4  ALGEBRA. 

usually  represents  both  the  arithmetical  value  and 
the  quality  of  an  algebraic  number.  Thus  a  may 
denote  +5,  —  5,  —  8,  +  17,  or  any  other  algebraic 
number.  When  no  sign  is  written  before  a  symbol 
of  number,  the  sign  +  is  understood. 

Known  Numbers,  or  those  whose  values  are  known, 
or  supposed  to  be  known,  are  denoted  by  figures,  or 
the  first  letters  of  the  alphabet,  as  ^,  b,  c,  a',  b\  c\ 
a  I,  b^,  c^, 

Uftknoivn  Numbers,  or  those  whose  values  are  to 
be  found,  are  usually  denoted  by  the  last  letters  of 
the  alphabet,  as,  x,  j,  z,  x',  j/',  z\  x^,  y^,  z^. 

Quantities  represented  by  letters  are  called  literal; 
those  represented  by  figures  are  called  numerical, 

7.  Signs  of  Operation.  The  signs,  +  (read  plus), 
—  (read  minus'),  X  (read  multiplied  by),  -^  (read 
divided  by),  are  used  in  algebra  to  denote  algebraic 
addition,  subtraction,  multiplication,  and  division, 
respectively.  The  use  of  the  signs  +  and  —  to  indi- 
cate operations  must  be  carefully  distinguished  from 
their  use  to  denote  quality.  In  the  literal  notation, 
multiplication  is  usually  denoted  by  writing  the  mul- 
tiplier after  the  multiplicand.  Thus,  a  b  —  a  X  b. 
Sometimes  a  period  is  used;  thus,  4  •  5  ==  4  X  5- 
Algebraic  division  is  often  denoted  by  a  vinculum ; 

thus  -^  —  a  -^  b. 
b 

8.  Signs  of  Relation  and  Abbreviation.  The  sign  of 
equality  is  —.     The  sign  of  identity  is  =.     The  sign 


DEFINITIONS   AND   NOTATION.  5 

of  inequality  is  >  or  <  ,  the  opening  being  toward 
the  greater  quantity. 

The  signs  of  aggregation  are  the  parentheses  (  ), 
the  brackets  [  ],  the  brace  {  },  the  vinculum  ', 
and  the  bar  |.  They  are  used  to  indicate  that  two 
or  more  parts  of  an  expression  are  to  be  taken  as  a 
whole.  Thus,  to  indicate  the  product  of  c  —  f/  multi- 
plied by  Xy  we  may  write  (c  —  d)  x,  \c—  d\  x,  {c  —  d\  x, 

c  —  dx,  or  —d\   . 

The  sign  .*.  is  read  heJtce,  or  therefore;  the  sign 
•.*  is  read  sijice^  or  because. 

The  sign  of  continuation  is  three  or  more  dots  ...  , 
or  dashes  —  ,  either  of  which  is  read  and  so  on. 

9.  The  result  obtained  by  multiplying  together 
two  or  more  numbers  is  called  a  Product.  Each 
of  the  numbers  which  multiplied  together  form  a 
product,  is  called  a  Factor  of  the  product. 

10.  A  Power  of  a  number  is  the  product  obtained 
by  taking  that  number  a  certain  number  of  times 
as  a  factor.  If  ;/  is  a  positive  integer,  a"  denotes 
aaaa  .,.\.o  n  factors,  or  the  «th  power  of  a.  In 
a!\  n  denotes  the  number  of  equal  factors  in  the 
power,  or  the  Degree  of  the  power,  and  is  called  an 
Exponent. 

11.  A  Root  of  a  quantity  is  one  of  the  equal  factors 
into  which  it  may  be  resolved. 

The  wth  root  of  a  is  denoted  by  '"{/a.  In  ^a, 
m  denotes  the  number  of  equal    factors  into  which 


6  ALGEBRA. 

^  is  to  be  resolved,  and  is  called  the  Index  of  the 
root.  The  sign  ^/~  (a  modification  of  r,  the  first 
letter  of  the  word  radix)  denotes  a  root.  If  no  in- 
dex is  written,  2   is  understood. 

12.  Any  combination  of  algebraic  symbols  which 
represents  a  number  is  called  an  Algebraic  Expression. 

13.  When  an  algebraic  expression  consists  of  two 
or  more  parts  connected  by  the  signs  +  or  — ,  each 
part   is  called  a  Term.      Thus,  the  expression 

^^  +  {c  —  X)  y  -\-  hz^  \  c  ^  d 

consists  of  four  terms. 

A  Monomial  is  an  algebraic  expression  of  one  term; 
a  Polynomial  is  one  of  two  or  more  terms.  A  poly- 
nomial of  two  terms  is  called  a  Binomial;  one  of 
three   terms   a  Trinomial. 

14.  The  Degree  of  a  term  is  the  number  of  its  lite- 
ral factors.  But  we  often  speak  of  the  degree  of  a 
term  with  regard  to  any  one  of  its  letters.  Thus, 
%cP'b'^x^,  which  is  of  the  ninth  degree,  is  of  the 
second  degree  in  a,  the  third  in  b,  and  the  fourth 
in  X. 

The  degree  of  a  polynomial  is  that  of  the  term  of 
the  highest  degree.  An  expression  is  homogeneous 
when  all  its  terms  are  of  the  same  degree. 

A  Liu'^ar  expression  rs  one  of  the  first  degree;  a 
Quadratic  expression  is  one  of  the  second  degree. 


DEFINITIONS   AND   NOTATION.  7 

15.  Any  algebraic  expression  that  depends  upon 
any  number,  as  x,  for  its  value  is  said  to  be  a  Function 
of  X.  Thus,  5  x"^  is  a  function  of  x ;  ^  x^  -{-  a^  —  y  x 
is  a  function  of  both  x  and  a;  but  if  we  wish  to  con- 
sider it  especially  with  reference  to  x,  we  may  call  it 
a  function  of  ;r  simply. 

A  Rational  Integral  Function  of  X  is  one  that  can 
be  put  in  the  form 

Ax"  +  Bx''-'^  +  Cx"-^  +  ...  +  F, 

in  which  n  is  a  whole  number,  and  A,B,  ...,  /^denote 
any  expressions  not  containing  x. 

Thus,  ax^— 4x^  —  dx  +  c  and x^  —  \x  are  rational  integral 
functions  of  x  of  the  third  degree. 

16.  The  Reciprocal  of  a  number  is  one  divided  by 
that  number. 

17.  If  a  term  be  resolved  into  two  factors,  ei- 
ther is  the  Coefficient  of  the  other.  The  coeffi- 
cient may  be  either  ninnerical  or  literal.  Thus,  in 
^abc^,  4  is  the  coefficient  of  abc^,  4a  of  bc^,  and 
4  a  d  of  c^.  When  no  numerical  coefficient  is 
written,  i  is  understood;  thus,  a  =  (^+  1^  a,  and 
-a  =  (-l}a. 

18.  Like  or  Similar  Terms  are  such  as  differ  only  in 
their  coefficients.  Thus,  4a bc^  and  loabc^  are  like 
terms  ;  6cP'l^y^  and  4ay^y^  are  like,  if  we  regard*6rt2 
and  4^  as  their  coefficients,  but  unlike  if  6  and  4  be 
taken  as  their  coefficients. 


8  ALGEBRA. 

19.  A  Theorem  is  a  proposition  to  be  proved. 

20.  A  Problem  is  something  to  be  done. 

21.  To  solve  a  problem  is  to  do  what  is  required. 

22.  An  Axiom  is  a  self-evident  truth. 

The  axioms  most  frequently  used  in  Algebra  are 
the  following: 

1.  Numbers  which  are  equal  to  the  same  number 

or  to  equal  numbers  are  equal  to  each  other. 

2.  If  the  same  number  or  equal  numbers  be  added 

to,  or  subtracted  from,  equal    numbers,   the 
results  will  be  equal. 

3.  If  equal   numbers  be   multiplied    by  the    same 

number  or  equal  numbers,  the  products  will 
be  equal. 

4.  If  equal  numbers  be  divided  by  the  same  num- 

ber, except  zero,   or    by   equal    numbers,   the 
quotients  will  be  equal. 

5.  Like  powers  or  like  roots  of  equal  numbers  are 

equal. 

23.  Identical  Expressions.      Equal  expressions  that 

contain    only  figures,  or  expressions    that  are  equal 

for    all   values    of  their    letters,   are    called    Identical 

Expressions. 

Thus,  4  +  6  and  5X2  are  identical  expressions;  so  also  are 
{a  ^b)  {a-  b)  and  a'  -  b\ 
0 

24.  An  Equality  is  a  statement  that  two  expressions 

represent  the  same  number.     The  two  expressions  are 
called  the  Members  of  the  equality. 


DEFINITIONS  AND   NOTATION.  9 

25.  Identities  and  Equations.  Equalities  are  of  two 
kinds,  identities  and  equations. 

The  statement  that  two  identical  expressions  are 

equal  is  called  an  Identity.     In  writing  identities,  the 

siy^n  =,  read  '  is  identical  with,'  is  often  used  instead 

of  the  sign  =. 

Thus  the  equality  5  +  7  =  4X3  is  an  identity  ;  so  also  is 
aP- —  x^  =■  {a  ■\-  x)  (a  —x),  since  it  holds  true  for  all  values  of  a 
and  X.  To  indicate  that  these  equalities  are  identities,  they  may 
be  written  $  +  7=  4^3  and  a^  —  x^=  (a  +  x)  {a  —  x). 

If  two  expressions  are  not  identical,  and  one  or 
both  of  them  contains  a  letter  or  letters,  the  state- 
ment that  they  are  equal  is  called  an  Equation. 

Thus  the  equalities  3,1-  — 6  =  0,  7^  =  2^  +  5,  and  2j  — 4:r  =  6, 
are  equations.  The  first  holds  true  for  x=  2,  and  the  second 
ior  a=  I.  The  equation  2j  — 4jr=6  holds  true  for^  =  2JI-  +  3; 
hence  if  :r  =  i ,  j  =  5  ;  if  ^-  =  2,  j  =  7 ;  and  so  on. 

26.  Algebra  is  that  branch  of  mathematics  which 
treats  of  the  equation,  its  nature,  the  methods  of  solv- 
ing it,  and  its  use  as  an  instrument  for  mathematical 
investigation. 

The  history  of  algebra  is  the  history  of  the  equation.  The 
notation  of  algebra,  including  symbols  of  operation,  rel  ition, 
abbreviation,  and  quantity,  was  invented  to  secure  conciseness, 
clearness,  and  facility  in  the  statement,  transformation,  and  solu- 
tion of  equations.  The  number  of  algebra,  which  has  quality  as 
well  as  arithmetical  value,  was  conceived  in  the  effort  to  interpret 
results  obtained  as  solutions  of  equations.  Hence  the  study  of 
the  nature  and  laws  of  algebraic  number  and  of  the  methods 
of  combining,  factoring,  and  transforming  algebraic  expressions 
should  be  pursued  as  auxiliary  to  the  study  of  the  equation. 
This  will  lend  interest  and  profit  to  what  might  otherwise  be 
regarded  as  dull  and  useless. 


10  ALGEBRA. 


CHAPTER   II. 
FUNDAMENTAL    OPERATIONS. 

27.  Addition  is  the  operation  of  finding  the  result 
when  two  or  more  numbers  are  united  into  one.  The 
result,  which  must  ahvays  be  expressed  in  the  sim- 
plest form,  is  called  the  Sum. 

28.  Subtraction  is  the  operation  of  taking  from  one 
number,  called  the  Minuend,  another  number,  called 
the  Subtrahend.  The  result,  which  must  be  expressed 
in  the  simplest  form,  is  called  the  Remainder. 

The  subtrahend  and   the  remainder  are  evidently 
the  two  parts  of  the  minuend;   hence,  since  the  whole 
is  equal  to  the  sum  of  all  its  parts,  we  have 
minuend  =  subtrahend  +  remainder. 

29.  To  multiply  one  number  by  another  is  to  treat 

the  first,  called  the  Multiplicand,  in  the  same  way  that 

we  would  treat   i   to   obtain  the  second,  called  the 

Multiplier. 

Thus,  3=1  +  1  + I  ;  .-.4x3  =  4  +  4  +  4. 
Again,  ^=i-+3  X  2;  .-.  9X  ^  =  9  +  3  X  2. 

30.  Having  given  a  product  and  one  factor.  Division 
is  the  operation  of  finding  the  other  factor.  The 
given  product  is  called  the  Dividend ;  the  given  fac- 
tor, the  Divisor;   and  the  required  factor,  the  Quotient. 

From  their  definitions,  the  divisor  and  quotient  are 
evidently  the  two   factors  of  the  dividend  ;    hence, 


FUNDAMENTAL  OPERATIONS.  II 

since  any  number  is  equal  to  the  product  of  its 
factors,  we  have 

quotient  X  divisor  =  dividend.  (i) 

Let  D  denote  the  dividend,  and  d  the  divisor ;  then 
the  expression  D  -^  d  will  denote  the  quotient,  and 
by  (i)  we  shall  have 

{D-.d)xd  =  D,  (2) 

31.  Law  of  Order  of  Terms.     Numbers  to  be  added 
may  be  arraiiged  in  any  order ;    that  is, 

a^b  =  b^ra.  (A) 

For  let  there  be  any  two  quantities  of  the  same  kind, 
one  containing  a  units  and  the  other  b  units.  Now  if 
we  put  the  second  quantity  with  the  first,  the  measure 
of  the  resulting  quantity  will  be  ^  +  <5  units ;  and  if  we 
put  the  first  quantity  with  the  second,  the  measure  of 
the  resulting  quantity  will  be  ^  +  ^  units.  It  is  self- 
evident  that  these  two  resulting  quantities  will  be 
equal;   hence  their  measures  will  be  equal; 

.' .  a  ^  b  =  b  ^  a. 
A  similar  proof  would  apply  to  an  expression  of  any 
number  of  terms. 

32.  Law  of  Grouping  of  Terms.     Numbers  to  be  added 
may  be  grouped  in  any  manner ;  that  is, 

a^^b^c^a^ib^c),  (B) 

For  by  the  laiv  of  order  we  have 
a-\-b^  c=b  ^c-\-  a 

=^  {b  -^  c)  ^  a  =  a  ^  {b  ^-  c). 

A  similar  proof  would  apply  in  any  other  case. 


12  ALGEBRA. 

33.  Law  of  Quality  in  Products.  Two  like  signs  give 
+ ;  tzvo  unlike  signs  give  — . 

By  the  definition  of  multiplication  we  have 

+  3-0+(+i)  +  (+i)4-(+i). 
.-.    (+4)x(+3)-+(+4)+(+4)  +  (-f4)-+i2,    (i) 
and 

(-4)  X  (+3)  =  +  (-4)  +  (-4)  +  (-4)  =-  12.    (2) 

Again, 

-3=0-(+i)-(+i)-(+i). 

•••    (+4)X(-3)=-(+4)-(+4)-(+4)=-i2,   (3) 
and 

(-4)x(-3)  — (-4)-(-4)-(-4)-4-i2.    (4) 

From  (i )  and  (4)  it  follows  that  two  factors  like  in 

quality  give  a  positive  product ;  and  from  (2)  and  (3) 

it  follows  that  two  factors  opposite  in  quality  give  a 

negative  product. 

Thus  the  product  ab  \^  positive  or  negative  according  as  a 
and  b  are  like  or  unlike  in  quality. 

34.  Cor.  I.  Any  product  containing  an  odd  number 
of  7iegative  factors  will  be  negative  ;  all  other  products 
will  be  positive. 

Hence,  changing  the  quality  of  an  even  number  of 
factors  will  not  affect  the  product;  but  changing  the 
quality  of  an  odd  number  of  factors  will  change  the 
quality  of  the  product. 

35.  Cor.  2.  The  quality  of  any  term  is  changed  by 
changing  the  quality  of  any  one  of  its  factors,  or  by 
multiplying  it  by  —  \. 

The  quality  of  any  expression  is  changed  by  changing 
the  sign  before  each  of  its  terms. 


FUNDAMENTAL   OPERATIONS.  13 

36.  Law  of  Order  of  Factors.  Factors  may  be  ar- 
ranged in  any  order ;    that  is, 

ab  =  ba.  (A') 

For  from  arithmetic  we  know  that  any  change  in  the 
order  of  factors  will  not  change  the  arithmetical  value 
of  their  product;  and  from  the  law  of  quality,  any 
change  in  the  order  of  factors  will  not  change  the 
quality  of  their  product. 

Thus,  (+  4)  (-  3)  (-  5)  =  (-  5)  (+  4)  (-  3)  =  (-  3)  (-  5)  (+  4). 

37.  Law  of  Grouping  of  Factors.  Factors  may  be 
grouped  in  any  manner  :  that  is, 

abc  =  a{bc),  (B') 

For  by  the  law  of  order  we  have 
abc  —  bca. 

=  {bc)  a  =  a  (b  c). 

Since  a  (bcd)=  bc(da),  a  product  is  multiplied  by 
any  number  a  by  multiplying  one  of  its  factors  by  a. 

Note.  The  laws  of  order  and  grouping  are  often  called  the 
commutative  and  associative  laws  of  addition  and  multiplication. 

38.  Equimultiples  of  two  or  more  expressions  are 
the  products  obtained  by  multiplying  each  of  them 
by  the  same  expression. 

Thus,  A  m  and  B  jn  are  equimultiples  of  A  and  B. 

39.  If  Am  =  Bm  (i) 

and  m  is  not  zeroy  then 

A  =  B.  (2) 

For  dividing  each  member  of  identity  (i)  by  m,  we 

obtain  identity  (2). 


14  ALGEBRA. 

40.  Distributive  Law.  The  product  of  two  expres- 
sions is  equal  to  the  sum  of  the  products  obtained  by 
multiplying  each  term  of  either  expression  by  the  other^ 
and  conversely.    That  is, 

{a-\-b-\-c-\- ,..)  x^ax^bx-\-cx-\-.,,,       (C) 

Let  m  and  n  denote  any  positive  integers^  and  a  and 
b  any  numbers  whatever;  then  we  have 

{a  -\-  b)  m^i{a  -{-  b)  -^  {a  ^  b)  ^  . ..  io  m  terms 
^^am  +  bm.  (i) 

Again, 

{a  +  b)  {ni  -^  n)^  a  {m  -=r  fi)  -\-  b{m-^  n).      (2) 

For  multiplying  each  member  of  (2)  by  n,  we  obtain 

the  identity  (i);   hence,  by  §  39,  (2)  is  an  identity. 

Hence, 

{a  +  b)z^az  +  bz,  (3) 

in  which  z  is  any  positive  number. 

Again, 

{a  J^b){-z)=a{-z)-\-b  (-  z),  (4) 

For  changing  the  quality  of  the  members  of  (4),  we 

obtain  identity  (3) ;   hence,  by  §  39,  (4)  is  an  identity. 

The  same  reasoning  would  apply  to  any  polynomial 

as  well  as  to  ^  +  ^/  hence  {C)  is  proved  for  all  values 

oi  X. 

By  this  law  similar  terms  are  united  into  one;  thus  3^;r 
—  y  bx-\-/^cx=  (■^a  —  yd  +  4c)x. 

41.  Law  of  Exponents.  Let  7n  and  n  be  any  positive 
integers,  then  by  definition  we  have 

a"' a"  =  (a  a  a  ...  m  factors)  {a  a  a  ...ton  factors) 
=  aaa  ...m  +n  factors  =  a"'+"- 


FUNDAMENTAL   OPERATIONS.  1$ 

42.  From  the  laws  (A),  (B),  (C)  of  §§  31,  32,  40, 
we  have  the  following  Rale  for  Addition : 

Write  the  expressions  iinder  each  other ^  so  that  like 
terms  shall  be  in  the  same  coltmm  ;  then  add  the  col- 
ufuns  separately. 

43.  Rule  for  Subtraction.  To  subtract  one  algebraic 
expression  from  another,  add  to  the  minuend  the  sub- 
trahend with  its  quality  chafigcd. 

For  let  5  denote  the  subtrahend,  and  R  the  re- 
mainder; then  {R  +  S)  will  denote  the  minuend,  and 
—  5  the  subtrahend  with  its  quality  changed.     But 

Hence,  parentheses  preceded  by  the  sign  —  may  he 
removed  if  the  sign  before  each  of  the  included  terms 
be  changed  fro7n  +  to  —  or  from  —  to  +. 

Thus,  a c  —  {fn  —  2 c  n  ^  "i^  a x)  —  a  c  —  m ^r  ic n  —  },a X. 

In  arithmetic  addition  implies  increase,  and  subtraction  de- 
crease ;  but  in  algebra  addition  may  cause  decrease,  and  sub- 
traction increase.  To  solve  any  problem  of  subtraction  in 
algebra,  we  first  reduce  it  to  one  of  addition. 

44.  Since  3^  —  4^  =  3^—  (4-4)<^  =  3^  +  (—  4)<^> 
we  evidently  may  regard  the  sign  connecting  two 
terms  as  a  sign  either  of  quality  or  of  operation.  In 
general  formulas,  it  is  of  advantage  to  regard  the 
sign  +  or  —  as  a  sign  of  operation ;  but  in  most 
other  cases  it  is  better  to  regard  the  sign  written 
before  any  term  as  the  sign  of  its  numerical  coeffi- 
cient, the  sign  of  addition  being  understood  between 
each  two  consecutive  terms. 


l6  ALGEBRA. 

45.  From  the  commutative,  associative,  and  dis- 
tributive laws  of  multiplication  we  have  the  three 
following  rules: 

1.  To    multiply  monomials   together,   multiply  to- 

gether tJieir  numerical  coefficients ,  observing 
the  law  of  sights ;  after  this  result  write  the 
product  of  the  literal  factor  Sy  observing  the  laiv 
of  exponents, 

2.  To  multiply  a  polynomial  by  a  monomial,  mul- 

tiply each  term  of  the  polynomial  by  the  mono- 
mial^ and  add  the  results. 
In  applying  the  law  of  signs,  each  term  must 
be  considered  as  having  the  sign  which  pre- 
cedes it. 

3.  To   multiply  one   polynomial  by  another,  mul- 

tiply the  multiplicand  by  each  term  of  the 
multiplier y  and  add  the  results  thus  obtained. 

Let  the  student  state  in  words  the  following  im- 
portant theorems: 


{a-^cy  =  a^\2ac^c^. 

(0 

{a  —  cY-a^—iac^c^. 

•          (2) 

(a^c){a.-c)=^a^-c\ 

(3) 

46.  Law  of  Quality  of  Quotient.  Like  signs  in 
dividend  and  divisor  give  +  in  the  quotient;  unlike 
signs  give  — . 

Let  d  =  divisor,  ^  =  quotient ;  then  g  d=  dividend. 
By  §  33  i^  ^  ^^^  Q^  have  like  signs,  q  must  be  +; 
while  if  d  and  q  d  have  unlike  signs,  q  must  be  — . 


FUNDAMENTAL  OPERATIONS.  1 7 

Hence  changing  the  quality  of  both  dividend  and 
divisor  does  not  affect  the  quotient ;  but  changing 
the  quahty  of  either  the  dividend  or  the  divisor 
changes  the  quahty  of  the  quotient. 

47.  AX   -  =  ^^  (I) 

m       m 

For  muhiplying  each  member  of  (i)  by  W  we  obtain 
A  =  A. 

By  (i),  ±_£=  (^ad)c-  =al^~'  §39. 

e  e  e 

That  is,  any  product  may  be  divided  by  any  number 
by  dividing  one  of  its  factors  by  that  number. 

48.  Let  D  =  the  dividend,  d  =  the  divisor,  and 
^  =  the  quotient;   then  D  =  dq. 

Hence,  by  §  §  23,  40,  47,  we  have 

m£>  =  d(mq)y   or   -=d^\  (i) 

mm 

JD={md)^,    or   D  =  ~(mq)',         (2) 
m  m  ^  "^ 

and  ml?=  (m  d)  q,    or    -  =  [^  a.  (t\ 

m        \mj  ^^^ 

From  equations  (i),  (2),  and  (3),  respectively,  it 
follows  that, 

(i.)  Multiplying  or  dividing  the  divideJid  by  any 
quantity  multiplies  or  divides  the  quotient 
by  the  same  quantity. 


1 8  ALGEBRA. 

(il.)  Multiplying  or  dividi7ig  the  divisor  by  any 
quantity  divides  or  multiplies  the  quotient 
by  the  same  quantity. 

(iii.)  Multiplying  or  dividing  both  dividend  and 
divisor  by  the  same  quantity  does  not  affect 
the  quotient. 

49.    Law   of  Exponents.     If  m   and  ;/  are   positive 
integers,  and  ;;/  >  Uy 

a"'       aaaa  ...to  m  factors 


a"        aaaa...  to  ;/  factors 

==  aa a  ...  to  m  —  n  factors  §  48. 

=  a—\  (i) 

50.  Corollary.  If  in  (i)  ;;^  =  n,  the  first  mem- 
ber is  evidently  i,  and  the  second  is  a^;  hence  a^=  i. 

That  is,  any  quantity  with  zero  as  an  exponent 
equals  unity. 

51.  The  Distributive  Law.  A  quotient  equals  the 
sum  of  the  quotients  of  the  parts  of  the  dividend  di- 
vided by  the  divisor. 

For  ^jLfpA^(a^c~d)\  §47- 

b  b 


FUNDAMENTAL  OPERATIONS.  I9 

52.  To  divide  one  monomial  by  another. 

(i.)  If  all  the  literal  factors  of  the  divisor  appear 
in  the  dividend,  divide  the  numerical  coeffi- 
cient of  the  dividend  by  that  of  the  divisor, 
observing  the  law  of  signs ;  then  divide  the 
literal  parts,  observing  the  law  of  expojietits 
(§§47.48). 
(ii.)  If  all  the  literal  factors  of  the  divisor  do  not 
appear  in  the  dividend,  cancel  all  the  factors 
common  to  both  the  dividend  and  divisor 

(§§47.48). 

53.  From  the  distributive  law  we  have  the  follow- 
ing two  rules: 

(i.)  To  divide  a  polynomial  by  a  monomial,  di- 
vide each  term  of  the  polynomial  by  the 
monomial  and  add  the  results. 
(ii.)  To  divide  one  polynomial  by  another,  ar- 
range both  dividend  and  divisor  according  to 
the  powers  of  some  letter.  Find  the  first  term 
of  the  quotient  by  dividing  the  first  term  of 
the  dividend  by  the  first  term  of  the  divisor. 
Multiply  the  divisor  by  the  term  thus  foimd, 
and  subtract  the  product  from  the  dividend. 
Treat  this  remainder  as  a  neiv  dividend  aiid 
repeat  the  process  until  there  is  no  remaiftder, 
or  until  a  remainder  is  found  ivhich  will  not 
contain  the  divisor.  Write  the  remainder 
over  the  divisor  as  a  part  of  the  qtwtie?tt. 


20  ALGEBRA. 

The  several  products  and  the  remainder  are  the 
parts  into  which  the  process  has  separated  the  divi- 
dend ;  and  the  quotient  found  is  made  up  of  the  quo- 
tients of  these  parts  divided  separately  by  the  divisor. 
Hence  by  the  distributive  law  it  is  the  quotient 
required. 

54.    Detached   CoeflBcients.      If  two  polynomials  in- 
volve but  one  letter,  or  are  homogeneous  and  involve 
but  two  letters,  much  labor  is  saved  in  finding  their 
product  or   quotient  by   writing  simply  their  coeffi- 
cients.    The  coefficient  of  any  missing  term  is  zero, 
and  must  be  written  in  order  with  the  others, 
(i)    Multiply  ^x^  +  2x^  -S  by  ^ir^  +  3  —  5  x. 
3+2+0-8 
I-    5+    3 


3+    2+    o    -8 
—  15  —  10    —0  +  40 

+    9    +6+    0-24 


3  -  13  -    I     -2  +  40-24 
Hence  the  product  is  ;^  x^  —  1$  x^  —  x^  —  2  x^  +  40  x  —  24. 
(2)    Divide  2  x^  -  S  x  +  x^  +  12  -  y  x^  by  x^  +  2  -  ^  x. 

1  +  2—    7—    8+i2|i  — 3  +  2 

■-3+  2  ,+5+6 


+  5- 

9- 

8 

+  5- 

15  + 

10 

+ 

6- 

18  + 

12 

+ 

6- 

18  + 

12 

Hence  the  quotient  is  jr^  +  5  :ir  +  6. 


FUNDAMENTAL   OPERATIONS.  21 


EXERCISE  I. 

1.  Find  by  multiplication  the  value  of  (x  +  jY,  {x  +/)*, 
(x  +  yy%  (x  +  y)%  {x  +  y)\  {X  +  7)«. 

Verify  by  division  or  multiplication,  and  fix  in  mind,  the 
following  identities : 

2.  If  «  is  any  positive  whole  number, 


x  —  y 

If  ^  =  I,  this  identity  becomes, 

3-    If  //  is  any  positive  even  number, 
x  -\-  y  "^  "^ 


4.    If;/  is  any  positive  odd  number, 

=  ;t:"-^  — ^"-2^  +  a'"-V ^y-24-y-i. 


^' +/'_,,_,      ^_2 


x  -\-  y 

5.    Show  that  x"  +  v"  is  not  exactly  divisible  by  ^  +  j;  or 
X  —  y,  when  /i  is  any  even  whole  mmiber. 


22  ALGEBRA. 


CHAPTER    III. 
FRACTIONS. 

55.  An  Algebraic  Fraction  is  the  indicated  quotient 
of  one  number  divided  by  another.  The  dividend  is 
called  the  Numerator,  and  the  divisor  the  Denominator 
of  the  fraction.  The  numerator  and  denominator  of 
a  fraction  are  called  its  terms.  In  fractions  division 
is  denoted  by  the  vinculum. 

Thus, ,  denotes  the  quotient  oi  a  —  b  divided 

c  ■\-  a 

by  c  +  d.     Here    the   vinculum    between    the    terms 

serves  as  a  sign  both  of  aggregation  and  of  division. 

56.  Law  of  Signs.  The  law  of  signs  in  fractions 
is  the  same  as  that  in  division.  The  sign  before  a 
fraction  is  the  sign  of  its  coefficient 

Thus,  -Zl^  :=  (_  i)IL_^  ^  ^.  §§  34,  46. 

0  b         0 

57.  An  Entire  or  Integral  Number  is  one  which  has 
no  fractional  part. 

A  Mixed  Number  is  one  which  has  both  an  entire 
and  a  fractional  part. 

58.  The  terms  Simple  Fraction,  Complex  Fraction^ 
Compound  Fraction,  and  Common  Denominator  are 
defined  in  algebra  as  in  arithmetic. 


FRACTIONS.  23 

59.  The  Lowest  Common  Denominator  of  two  or 
more  literal  fractipns  is  the  expression  of  lowest  de- 
gree that  is  exactly  divisible  by  the  denominators 
of  each  of  the  fractions. 

60.  To  reduce  a  fraction  to  an  equivalent  entire  or 
mixed  number,  perform,  in  whole  or  in  party  the 
indicated  operation  of  division. 

61.  To  reduce  a  mixed  number  to  an  equivalent 
fraction,  vinltiply  the  entire  part  by  the  denominatory 
to  the  product  add  the  numerator^  and  nnder  the  sum 
write  the  denominator  (^^^  22,  51). 

62.  To  reduce  a  fraction  to  its  lowest  terms,  can- 
cel all  the  factors  common  to  both  the  Jiumenitor  and 
the  denominator  (§  48). 

63.  To  reduce  fractions  to  a  common  denominator, 
multiply  both  terms  of  each  fraction  by  the  denomina- 
tors of  all  the  other  fractions.  Or^  find  the  lowest 
common  denominator  of  the  givcji  fractions.  Then 
multiply  both  terms  of  each  fraction  by  the  quotient  of 
the  lowest  common  denominator  divided  by  the  denom- 
inator of  that  fraction. 

This  operation  will  not  change  the  value  of  the 
fraction  (§  48),  and  the  resulting  fractions  will  evi- 
dently have  a  common  denominator. 

64.  To  add  or  subtract  fractions,  reduce  them  to  a 
commofi  denominator,  add  or  subtract  their  numera- 


24  ALGEBRA. 

tors^  and  place  the  sum  or  remainder  over  the  common 
denominator  (§§  48,  51). 

65.  To  multiply  a  fraction  by  an  entire  number, 
multiply  the  numerator  or  divide  the  denominator  by 
the  entire  number  (§  48). 

66.  To  divide  a  fraction  by  an  entire  number, 
divide  the  numerator  or  multiply  the  denominator  hy 
the  entire  number  (§  48). 

"•  |x^=i^X^  §§^3,65,66. 


a  c 
Yd 


§65, 


Hence  the  product  of  two  or  more  fractions  equals 
the  prodtict  of  their  numerators  divided  by  the  product 
of  their  denomitiators. 

6a  \^'^^^c  §§48,65. 

bat 

be 

Hence  the  quotient  of  one  fraction  divided  by  an- 
other  equals  the  pr^oduct  of  the  first  multiplied  by  the 
second  invei'ted. 

_  i 

69.   Corollary.     Since  -  -^  ^  =  -.  the  reciprocal 

I       b      a  ^ 

of  a  fraction  equals  the  fraction  inverted. 


THEORY   OF  EXPONENTS.  2$ 


CHAPTER  IV. 

THEORY   OF   EXPONENTS. 

70.    U  a,  b,  in,  11,  denote  any  numbers,  the  five  laws 
of  exponents  may  be  expressed  as  follows : 


a"'  X  a"  =  a'"  + ". 

(i) 

a" 

(2) 

{ary  =  a"'\ 

(3) 

{abY  =  a'"b'\ 

(4) 

\b)          b- 

(s) 

These  laws  hold  for  any  exponents,  whether  they 
be  integral,  fractional,  positive,  or  negative. 

71.    To  prove  the  five  laivs,  when  the  exponents  m 
and  n  are  positive  integers. 

(i.)  Law  (i)  is  proved  in  §  41. 

(ii.)  Law  (2)  is  proved  in  §  49,  when  w  >  «  or  w  =  «. 

(iii.)  {a'")"  =  a""  a'"  ...to  u  factors  §  10. 

__  ^,«  +  ,«  +  ...  to  «  terms  La^   ^  j  ^ 

=  a"'". 


26  ALGEBRA. 

(iv.)  iaby  =^ab'ab-ab...\.om  factors  §  lo. 

—  {aa..,X.o  tn  factors)  {b  b  ...iom factors) 
=  ^-^-.  §§  38,  39- 

^"^KV    =  J  7/^ -to  ^factors  §10. 


aa a-"  io  m  factors 
bb b  -"io  ?n  factors 


§67. 


72.  A  Positive  Fractional  Exponent  denotes  a  root 
of  a  power.     The  denominator  indicates  the  root,  and 

r  

the  numerator  the  power;   that  is,  a""  —  ^oT. 

73.  Let  r  and  s  be  any  positive  whole  numbers, 

and  let  \'a  —  c^  or  a  —  c^  \ 

then  {'S/^^Y  =  ^^ 

and  a^  =  (O'  =  ^"^  =  (f  0'. 

...  ^^  =  c''=  ('{/ay. 

r  

Hence  a'  denotes  either  v^a'  or  its  equal  (v^a)^ 

74.  Negative    Exponents.      If  we   assume   law  (2), 
§  70,  to  hold  when  m  —  o,  we  have 

or 

That  is,  a~ "  denotes  the  reciprocal  of  a". 

75.  To  prove  the  five  laws,  when  the  exponents  m 
and  n  are  positive  fractions. 


THEORY   OF  EXPONENTS.  2/ 

(i.)  Let  /,  q,  r,  s,    denote   any   positive   integers ;   then 
by  §  73  we  have 

09  a?  =  («^  «''...  to  /  factors)  (a'  a^  .--to  r  factors), 

and      a^    •'"  =  (a'^  a'^  -- -  to />  factors)  (a'  a^  -"to  r  factors). 
/!  r       l^- 

9  n'  =  /7^      •'- 


Cf  4..     I 


a*  a"  =  a 

p_ 

(ii.)       -^^^^'x    _ 

a'  a' 

t     -L         >*_♦' 

(iii.)    \a'0  =  a^  '  a^  -"  to  r  factors 

^  +  -^  4- ...  to  r  terms 

=  a"    ^ 

=  «^. 

...     {a'y={a'^y  §§23,72. 

=  [U')'t  §  72. 

Now  one  of  the  s  equal  factors  of  one  of  the  q  equal  fac- 
tors of  any  number  is  evidently  one  of  the  q  s  equal  factors 
of  that  number ;  that  is, 

(iv.)  (^1^)^=^^"^ 

=  [(f  b'  -a'b'  ...  to  s  factors j"     §  §  ^Z,  39. 
I   I 


28  ALGEBRA. 

r     r 

=  a"  b\ 
(v.)  Let        |  =  r,  or  a  =  b c; 


§  73' 


then 


g^'^ 


and  a'  =  {bey  =  b' c%  or  ~  =  c' 


(I)--? 


If 

p.r  rp 


Corollary.    By  (i),  ^^  =  GO"  =  a^'. 

76.      To  prove  the  five  laws,  when  the  exponents  arc 
negative. 

Let  h  and  k  be  any  positive  numbers. 


(i.)  .-«-  =  -^X^  §74. 

§§67,    71. 


/j  +  /fe 


« 


=  ^-<^'  +  ^)  =  ^-^-^ 


74. 


^"•^  ^^  =  ^-^  =  ^-'-'  §48. 


THEORY   OF  EXPONENTS.                         29 

(iii.)  («-'■)-*= I  ^(^y 

§  74. 

=  ■:;.=«-. 

§§71,68. 

<'^->  (''")-^=(.V 

§74. 

I         II 

-*^-*. 

<-)  .©"'=-(1)' 

_     .  «*  _  ^*  «-*  <^-*  _  «-* 
~  ^  ^  ^  ~  ^  *  ^^ '  /^-'^'  ~  ^'  * 

Note.  The  introduction  of  fractional  and  negative  expo- 
nents is  evidently  not  necessary  ;  but  they  supply  us  with  a  new 
notation  of  very  great  convenience. 

77.  If  we  use  the  term  power  to  signify  what  is  in- 
dicated by  any  exponent,  the  five  laws  of  exponents 
may  be  stated  as  follows : 

(i.)  The  product  of  the  vnth  and  the  nth  power  of 
any  number  equals  the  (m  +  n)///  power  of 
that  number, 

(ii.)  The  quotient  of  the  vath  power  of  any  number 
divided  by  its  nth  power  equals  the  (m  — n)/// 
power  of  that  number. 

(iii.)  The  v\th  power  of  the  vs\th  powei"  of  any  number 
equals  the  m  vith  pozver  of  that  number. 


30  ALGEBRA. 

(iv.)  The  mth  power  of  the  product  of  any  niiniher  of 
factors  equals  the  product  of  the  vnth  powers 
of  those  factors. 

Corollary.  The  rth  root  of  the  product  of  two 
or  more  factors  equals  the  product  of  their  ^th  roots. 

(v.)  The  vnth  power  of  the  quotient  of  any  two  quan- 
tities equals  the  quotient  of  their  mth  powers. 

Corollary.  The  rth  root  of  the  quotient  of  any 
two  numbers  equals  the  quotient  of  their  rth  roots. 

78.  To  affect  a  monomial  product  with  a  given 
exponent,  multiply  the  exponent  of  each  factor  by  the 
given  exponent. 

This  rule  follows  from  laws  (3)  and  (4). 

Thus,      {^a'b-^c'^)^  =  4^  {J)"^  (/^  -  ^)^  {c^^ 

^_    a""     b^    ar^  _  _b^ 

Hence  a  factor  7nay  be  changed  from  one  term  of 
a  fraction  to  the  other  if  the  sign  of  its  expojient  be 
changed. 

EXERCISE  2. 

I.  Multiply  3  a^  b^  c"^  by  2  a^  b*  c^ ;  7  ^2  ^  -«^  - 1  by 
6  a^  x'"  ~  "7^". 


THEORY   OF  EXPONENTS.  3 1 

2.  Perform  the  operations  indicated  by  the  exponents 
in  each  of  the  following  expressions  :  \2  a"^  x~  ^y^J  ; 
(125  Jx-^^  ■       (8  a^  b^  c'  d-^)  ~ ^  ;       (64  «-  ^  a:~  ^)5 ; 

3.  In  each  of  the  following  expressions  introduce  the  co- 
efficient within  the  parentheses:  8  {a^  —  x"^)-  ;  a^  {a -\-  a^xy ; 
x^{i-  x^)^  ;  x^  (a  -  xj  ;  x''  (x''  -  ay)  ~  ^. 

(8)1  =  (8)3 '1  =  (8^)^  =  4! 

.-.    8  (^2  -  X^)i  =  4^  («2  _  ^2)1  =  (4  «2  -  4  ;r2)t. 


{•-if  i-^i?  (-3 

4.   Simplify  — ~r^ —  ; —  ;    - 


,2\| 


/    ,.vA!       (x^±r\^      (£!±2^      (x^+y^)^  ^ 

\    "*"  xy     _[     x'     )     _  _(£:)!__  -i-^  (x^-hy^)^ 


5.  Remove  a  monomial  factor  from  within  the  parentheses 
in  each  of  the  following  expressions  :  3  (d'^  —  a^  b'^)  "2  ; 
2  {()a''b-  i^ab^)^',     f(2  7^^^^-54««/^*)^. 


32  ALGEBRA. 

7.  Square    the    following   binomials:      b''  x~'"^  ~  a^  x" ; 

8.  Free  of  negative  exponents 

a-^  b^  ,         Sx~^y~i      .     g  x~^y~^  2-'^ 


3  :r       X  —  I  x^-\-  y 

r  "                              ^              0             a 

23  y                    x^ — y^ 

6                       3  2         /       ;<; 


10.    Simplify 


^    

I  :v 

I  + ^ 

I  2  jr'^ 

I  +  -  I  +^4- 

X  I  —  X 


a  +  b      a—  b      a^  +  ^  a'^x  -\-  ^  ax^  -}-  x^ 


^.^     c  -\-  d       c  —  d  x^  —  y^ 


c  —  d       c  •{■  d  x^  +  xy  +  y'^ 

2.   Simplify    (^  +  1^  +  ?)  -  f^+li'  -  ^-). 

,   Simplify    (^  +  i).(5-i  +  ^). 


FACTORING.  33 


CHAPTER  V. 

FACTORING,  HIGHEST  COMMON  DIVISOR,  LOWEST 
COMMON    MULTIPLE. 

80.  Factoring  is  the  operation  of  finding  the  fac- 
tors of  a  given  product.  It  is  the  converse  of 
multipHcation. 

81.  When  each  term  of  an  expression  contains 
the  same  factor,  the  expression  is  divisible  by  that 
factor. 

Thus,    x'^y  +  xy^  +  ^ax  -  zdx  =  x (xy  -i- y^  -\-  4a  ~  36). 
Also,    ac(ac  +  d)  +  d(ac  +  d)  =  {ac  +  d)(ac  +  d). 

Binomials. 

82.  Whatever  be  the  values  of  m  and  n, 

That  is,  t/ie  diffcrmce  between  any  two  quantities  is 
equal  to  the  product  of  the  sum  and  difference  of  their 
square  roots. 

Thus,  a^  -  x^-  (a*  +  ^'*)  (a*  -  x^). 


34  ALGEBRA. 

83.  From  the  examples  of  Exercise  i,  page  21,  we 
have, 

(i.)  x''  ~ y"  is  divisible  by  x  —  y  \{  n  is  any  whole 
number;  and  the  ;/  terms  in  the  quotient 
are  all  positive. 

(ii.)  x'' —  y"  is  divisible  by  x  +  y  \(  n  is  even; 
and  the  n  terms  in  the  quotient  are  alter- 
nately  positive  and  negative. 

Thus,     c^-b^^  («3 )'  _  (^i)* 

=  («3  _|.  ^i)  (^  _  (^  ^5  +  ^¥  4  -  ^5 ). 

(iii.)  x""  +y'  is  divisible  by  x  +  y  \i n  is  odd;  and 
the  n  terms  in  the  quotient  are  alternately 
positive  and  negative. 

Thus,     ^fio  +  j5  =  (;t-2  +  y)  {x'^  -  x^y  +  :r ^^  -  -^V^  +  JJ'*)  • 

(iv.)  ;r"+j/"  is  not  divisible  hy  x  ^  y  or  x  —  y 
when  ?2  is  even. 

84.  For  any  value  of  ;/  we  have, 

For  71  =  I,  (i)  becomes 

a:*  +  y  =  (a;2  +  /  +  ^jj;  /y/a)  (a;^  +y^  —  xy  ^fi). 


factoring.  35 

Trinomials. 

85.  x^  ±  2  ax  -h  a^  =  (x  ±  ay. 

That  is,  if  tivo  tenns  of  a  trinomial  are  positive,  and 
the  third  is  ±  twice  their  square  roots ^  the  trinomial 
equals  the  square  of  the  sum  or  difference  of  the  two 
square  roots. 

86.  x^  +  {a  +  b)x  -^  ab  =  {x  -\-  a)  {x  -[-  b) ; 
hence  x"^  -\-  c  x  -\-  d  =  {x  +  a)  {x  +  ^), 

if  a  -\-  b  =  c  and  ab  =  d,  (i) 

Equations  (i)  can  always  be  solved  for  a  and  b  by 
the  method  of  §  i66;  hence  a  trinomial  of  the  form 
x'^  -\-  ex  ■{■  d  can  be  resolved  into  two  linear  factors 
in  X.  Equations  (i)  however  may  often  be  solved 
by  inspection. 

Example.     Factor  x^y^  —  1 1  x^y^  +  30. 

^6^,4  _  1 1  x^y2  +  30  =  {x^y^y^  -  1 1  (:r3y)  +  30; 
hence  a  +  b  =  —  11,  and  «  ^  =  30  ; 

therefore  ^z=—  5,  b  =  — 6; 

whence     x^y*'  -  1 1  x^y^  +  30  =  (x^y'^  -  5)  (x^y"^  -  6). 

87.  nx^  +  ex  +  d= — ' ^^ ^—^ 

Now  (^nxy  -f-  c(ux)  +  nd  can  be  factored  by  §  86. 
Hence  any  trinomial  of  the  form  nx"^  +  ex  -\-  d  can 
be  resolved  into  two  linear  factors  in  x. 


^6  ALGEBRA. 

Example.     Factor  15  Jt:^  —  7  jc  —  2. 

15 

(i^x  ~  10)  (15  X  -\-  2)       /  \  /■         ,      \ 

88.  When,  by  increasing  one  of  its  terms,  a  trino- 
mial can  be  made  a  perfect  square,  it  can  be  factored 
by  §  82. 

Example.     Factor  x*  —  ^  a^  x"^  +  g  a^. 

x^-  Za^^'^  +  ga^  =  ^^  +  6a^x^  +  ga^~ga^x^ 
=  {x^+  3  a'^f  -  (3  a^  xf 
=  (;r2  +  3  «*  +  3  a'^x){x'^  +  3  «*  -  3  a'^x). 

89.  A  polynomial  of  four  or  more  terms  may 
often  be  factored  by  properly  arranging  its  terms, 
and  applying  the  foregoing  principles. 

1.     cx"^  —  cy'^  —  ax'^  +  ay'^  =  c  (x'^  —j")  —  ^  (-^^  —y^) 
=  (c-a)  (x-y)  (x+y). 

3.    x*-x'^~g-2a^x^+a^+6x  =  (x^  -  a'^f  -  {x  -  3)2 

=  {x^-a^+x-  3){x^-  a'^-x  +  3). 

EXERCISE  3. 

Resolve  into  their  simplest  factors  : 

1.  a^c^-\-acd-\-abc-\-bd. 

2.  a'-y^  -  Py  x""  -  a"  dy''  +  ^'  dx\ 


FACTORING.  37 

3.  10^^  +  30  x^ y  —  8  x)p-  —  24^. 

4.  {a^bf-i.  II.    2X'-<,xy  ^  2>y''' 

5.  d^b^  —  T^abc  —  \0(^.       12.    12  jc^  —  23  ;c^  +  5^. 

6.  98  —  7  jc  —  j(;-.  13.    9  x^  +  24xy  +  i6y\ 

7.  ^''  +  I.  14.   ^^  +  i6jc2  +  256. 

8.  ^«  -  I.  15.   81  a'  +  9  ^'  b'  +  b\ 

9.  7+10^  +  3:1:^  16.    2  +  7^—  ^S-^^- 
io.  6  ;c^  +  7  :x:  —  3.                   17-3  ^'  +  41  ^  +  26. 

18.  31^  —  35  —  6^=^. 

19.  a^  ^rb'^-c^-  d^  +  2  ^2/^2  _  2  ^2^2^ 

20.  I  —  «^  jc^  —  ^^_y-  +  2  rtt  <^  :i'_y. 

21.  ^^  JC  —  ^'"^JC  +  «^7  —  ^^j. 

22.  Resolve  A:^_y  —  jc^  j^^  —  x^ y-  -\-  x y^  into  four  factors. 

23.  Resolve  c^  —  64  a^  —  ^z^  +  64  into  six  factors. 

24.  Resolve   a,  {a  b  ^  c  df  -  {a^  +  ^,2  _  ^  _  ^2y  -^^^^ 
four  factors. 

25.  Write  out  the  following  quotients  : 

{x^-yh  -  {^^-yh> 

(x^  +  J)  -^  {x^  +  a^). 


38  ALGEBRA. 

HIGHEST    COMMON    DIVISOR. 

90.  A  Common  Divisor  of  two  or  more  expressions 
is  an  expression  that  divides  each  of  them  exactly. 
Two  expressions  are  prime  to  each  other  if  they  have 
no  common  factor  other  than  unity. 

91.  The  Highest  Common  Divisor  of  two  or  more 
algebraic  expressions  is  the  expression  of  highest 
degree  that  will  divide  each  of  them 'exactly. 

The  abbreviation  H.  C.  D.  is  often  used  for  the 
words  highest  common  divisor. 

92.  When  the  given  expressions  can  be  resolved 
into  their  simple  factors,  or  such  as  are  prime  to 
each  other,  their  H.  C.  D.  is  obtained  by  taking  the 
product  of  all  their  common  factors,  each  being 
raised  to  the  lowest  power  in  which  it  occurs  in 
any  of  the  expressions. 

Thus,  the  H.  C.  D.  of  6  (:r-i)  (;r+2)3  and  3  (;r-i)2  (:r  +  2)2 
{x  -  3)  is  zipc-  i)(;r+  2)2, 

93.  When  the  given  expressions  cannot  be  resolved 
into  their  factors,  the  method  of  finding  their  H.  C.  D. 
is  based  on  the  following  theorem : 

If  A  ^  B  Q  +  R,  then  the  H.  C.  D.  of  A  and  B 
is  the  same  as  the  H.  CD.  of  B  and  R, 

Since  R  —  A  —  B  Q,  by  §81  every  factor  com- 
mon  to  A     and  B    divides  R ;    hence    every   factor 


HIGHEST  COMMON  DIVISOR.  39 

common  to  A  and  B  is  common  to  B  and  R.  Con- 
versely, since  A  —  B  Q  +  R  every  factor  common 
to  B  and  R  divides  A  ;  hence  every  factor  common 
to  B  and  R  is  common  to  A  and  B. 

Hence  the  H.  C.  D.  of  i>'  and  R  is  the  H.  C.  D.  of 
A  and  /?. 

94.    To  find  the  //.  C  D.  of  two  algebraic  quantities. 

Let  A  and  ^  denote  any  two  rational  integral  func- 
tions of  X,  whose  H.  C.  D.  is  required,  the  degree  of 
B  not  being  greater  than  that  oi  A. 

Divide  A  by  B  aitd  let  the  quotient  be  Q^  and  the 
remainder  R^.  Divide  B  by  /?,  and  let  the  quotient  be 
Q2  and  the  remaiftdcr  R^,.  Divide  R^  by  R^  and  let 
the  quotient  be  (2..,  ^'^^  ^^^^  remainder  R^.  Continue 
this  process  until  the  remainder  is  zerOy  or  does  not 
contaift  x.  If  the  last  remainder  is  zero,  the  last  divi- 
sor is  the  H.  C.  D.  ;  if  the  last  rejnaiitdcr  is  not  zero^ 
there  is  no  H.  C.  D. 

From  the  process  above  described,  it  follows  that 

A  =  B  Q,^  R^, 
B  =  R,Q,  +  R,, 

^1=  ^2  a +  ^3. 


Now  by  §  93  the  pairs  of  expressions,  A  and  B, 
B  and  7?,,  R,  and  R^,  ...,  R  ,,-2  and  R,,_^,  all  have 
the  same  H.  C.  D. 


40  ALGEBRA. 

(i.)  If  R„  =  0,  R„_2  =  i?,,_i  g,.  Hence  the 
H.  C.  D.  o{  R^_,  and  7?„_„  or  R,^_,  a,  is 
y?„_i.  Hence  ie«_i  is  the  H.  C.  D.  oi  A 
and  ^. 

(ii.)  If  R„  is  not  zero,  the  H.  C.  D.  of  ^  and  B  is 
the  H.  C.  D.  of  R,,_^  and  R,  (§  93)-  But, 
since  R^  does  not  contain  x,  Rn-\  and  R„ 
have  no  common  factor  in  x.  Hence  A 
and  B  have  no  common  divisor. 

95.  Corollary.  To  avoid  fractions,  and  to  other- 
wise simpHfy  the  work  in  finding  the  H.  C.  D.,  it  is 
important  to  note  that  at  any  stage  of  the  process, 

(i.)  We  may  multiply  either  the  dividend  or  the 
divisor  by  any  quantity  that  is  not  a  factor 
of  the  other. 

(ii.)  We  may  remove  from  either  the  dividend  or 
the  divisor  any  factor  that  is  not  common 
to  both. 

(iii.)  We  may  remove  from  both  the  dividend  and 
the  divisor  any  common  factor,  provided  it 
is  reserved  as  a  factor  of  the  H.  C.  D. 

96.  To  find  the  H.  C.  D.  of  three  expressions, 
A,  B,  C,  find  the  H.  C.  D.  of  A  and  B,  and  then 
find  the  H.  C.  D.  of  this  result  and  C.  This  last 
H.  C.  D.  will  be  the  H.  C.  D.  of  A,  B,  and  C, 


LOWEST   COMMON   MULTIPLE.  4I 

LOWEST   COMMON  MULTIPLE. 

97.  A  Common  Multiple  of  two  or  more  expressions 
is  an  expression  that  is  exactly  divisible  by  each  of 
them. 

The  Lowest  Common  Multiple  (abbreviated  L.  C.  M.) 
of  two  or  more  expressions  is  the  expression  of  low- 
est degree  that  is  exactly  divisible  by  each  of  them. 

98.  Hence  when  two  or  more  expressions  can  be 
resolved  into  their  factors  the  L.  C.  M.  of  these  ex- 
pressions is  the  product  of  their  factors,  each  being 
raised  to  the  highest  power  in  which  it  occurs  in  any 
of  the  expressions. 

99.  To  find  the  L.  C.  M.  of  two  expressions,  as  A 
and  B,  when  they  cannot  be  factored,  divide  A  by  the 
H.  CD.  of  A  and  D  and  multiply  the  quotient  by  B. 

For  the  L.  C.  M.  of  A  and  B  must  evidently  con- 
tain all  the  factors  of  ^,  and  in  addition  all  the  factors 
of -^  not  common  to  A  and  B;  hence  the  rule. 

EXERCISE  4. 
Find  the  H.  C.  D.  of  the  following  expressions  : 

1.  x^  ■{■  2x^ —  Zx  —  id,  x^ -\- 2iX' —  2>x  —  2/^. 

2.  2  x^  —  2  x^  -\-  X'  -\-  -^  X  —  d,  4  ^*  —  2  ^^  +  3  ^  —  9. 

3.  4^^+I4^^+20Ar3+70^^  %X^^-2Zx^'—'^0(^—\2X^^^(iX^, 

Find  the  L.  C.  M.  of  the  following  expressions : 

4.  x^  -^  a  x^  ■\-  a^  X  ■\-  a*,   x*  -{•  a^  x^  +  a^. 

5.  x^  —  <)  x^  -f  26  ^  —  24,   x^  —  12  x^  ■\-  ^'J  X  —  60. 


42  ALGEBRA. 


CHAPTER  VI. 
INVOLUTION,  EVOLUTION,  SURDS,  IMAGINARIES. 

100.  Involution  is  the  operation  of  finding  a  power 
of  a  number. 

Evolution  is  the  operation  of  finding  a  root  of  a 
number. 

For  the  involution  and  evolution  of  monomials  see 
§  78,  of  binomials  see  §  275. 

101.  A  root  is  said  to  be  even  or  odd  according  as 
its  index  is  even  or  odd. 

By  the  law  of  signs  it  follows  that, 

(i.)   Any  odd  root  of  a  quantity  has  the  same  sign 
as  the  quantity  itself. 

(ii.)  Any  even  root  of  a  positive  quantity  may  be 
either  positive  or  negative.  In  this  chapter 
only  positive  even  roots  are  considered. 

(iii.)  Any  even  root  of  a  negative  quantity  is  not 
found  in  the  series  of  algebraic  numbers 
thus  far  considered. 

An  even  root  of  a  negative  number  is  called  an 
Imaginary  number.  For  sake  of  distinction  all  other 
numbers  are  called  ReaL 


EVOLUTION.  43 

102.    To  find  the  square  root  of  any  ntnnber. 

The  rule  is  given  by  the  formula, 

{a  +  hf  =  a""  -\-  {2  a  +  b)  b, 
in    which  a  represents  the  first  term  of  the  root  or 
the  part  of  the  root  already  found  ;  b  the  next  term 
of  the  root  ;  2  a  the  trial  divisor  in  obtaining  b;  and 
2  a  -]r  b  the  true  divisor. 

In  finding  the  square  root  of  any  polynomial,  as 
4  a:^  4-  py  +  13  x^y'^  —  6  xy^  —  4  x^y,  its  terms  should 
be  arranged  according  to  the  descending  powers  of 
some  letter,  and  the  work  may  be  arranged  as  below : 

I  2x^-xy+'iy^ 


4x*-  4x^y  +  1 3 x^y^  -  6xy^  +  9J/* 
4^ 


2a  +  b  =  4x^  —  xy 

{2a  +  b)b  = 


-4jr8j/+l3^V 
—  4x^y  +      x^y^ 


2a  +  b=  4x^  -  2xy  +  3 j/2 

{2a  +  b)b  = 


1 2  x^y^  —  6  xy^  +  gy* 
I2.r2y  -6xy^-\-gy* 


At  first  a=2x^  and   b  =  —xy;    then  a  =  2x^  —  xy 

and  b  =  3/^ 

The  root  is  placed  above  the  number  for  conven- 
ience. In  extracting  the  square  root  of  any  number 
expressed  in  the  decimal  notation,  we  first  divide  it 
into  periods  of  two  figures  each,  beginning  with 
units'  place.  We  then  proceed  essentially  as  with  the 
polynomial  above,  bearing  in  mind  that  a  denotes 
tens  with   reference  to  b. 


44 


ALGEBRA. 


103.    To  find  the  cube  root  of  any  number. 

The  rule  is  given  by  the  formula, 

{a  ^by  =  a^^{zd'^iab\  b'')b, 

in  which  a  represents  the  first  term  of  the  root  or  the 
part  of  the  root  already  found ;  b  the  next  term  of 
the  root;  ^  (f  the  trial  divisor  in  obtaining  b;  and 
3  ^^'-^  +  3  ^^^  +  <^^  the  true  divisor. 

In  finding  the  cube  root  of  any  polynomial,  as 
8  ;ir^  —  36  ;r^  +  66  ;i:4  +  l  —  63  ;r3  —  9  ;ir  +  33  ;t^,  its 
terms  should  be  arranged  according  to  the  descend- 
ing  powers   of  some   letter,   and   the  work   may   be 

arranged  as  below: 

I  2;i-2  — 3;r+  I 


a"  — 
,2 


8  :r6  -  36  ;jr5  +  66  Ji'i  -  63 -r^  +  33  ;i-2  -  9  ;r  +  I 


(3«'^  +  3^^  +  ^2)^  = 


■  ^6x^  +  66  z^  — 6s  :*^ 

36;r^+54:r^  — 27jr3 


3^2—  12X^~S^X^+27X^ 

Sab  +  d^=  6x^—gx+  I 

{3a'  +  3ab+b^)b  = 


I2x^  —  26x^  +  22x-~gx+i 
1 2  jr*  —  36  jr^  +  33  :i'2  —  9  jf  +  I 


At  first  a  =  2x^  and   b  =  —^xj    then  a  =  2x^  —  ^x  and 


In  finding  the  cube  root  of  any  number  expressed 
in  the  decimal  notation,  we  first  divide  it  into  periods 
of  three  figures  each,  beginning  with  units'  place. 
We  then  proceed  essentially  as  with  the  polynomial 
above,  a  denoting  tens  with  reference  to  b. 


EVOLUTION.  45 

104.  In  finding  the  fourth  root  of  any  number,  we 
may  obtain  the  square  root  of  its  square  root,  or 
follow  the  rule  given  by  the  formula. 

The  rule  for  finding  the  fifth  root  is  given    by  the 
formula, 

(^  +  ^)6  =  ^«  +  (5^^+  voa^b  ^  10^-^2^  5^^'  +  b')b. 

The  sixth  root  may  be  obtained  by  finding  the  cube 
root  of  its  square  root,  or  by  using  the  formula, 

In  like  manner  we  may  obtain  any  root  of  a  quantity. 

EXERCISE  5. 

Find  the  square  root  of 

1.  25  ^*  —  30^;^'*  +  49  ^^^'^  —  24^^^  +  16  a*. 

2.  ()  x'^  —  \2  X*  -\-   22  X^  -{■  X^  -\-   \2  X  -\-  /^, 

3.  384524.01.  4.     0.24373969. 

Find  the  cube  root  of 

5.  I  —  6  X  +  2 1  ^'-^  —  44  ^^  +  63  Jt:*  —  54  ^^  +  2*1  x^. 

6.  2d,x^y^-\-()(iX^ y^  —  (iX^y-{-x^—()(ixy^-\-(i/i^y^—t^(iX^y^. 

7.  3241792.  8.    191. 102976. 

9.    Find  the  fifth  root  of 

32  a:^  —  80.%*  4-  8o.;c^  —  40Jt;^  +  lo.:*:  —  i. 

10.    Find  the  fourth  root  of 

16^*  —  96  ^^jt:  +  216  a^x'^  —  216  «^^  +  81  .;c*. 


46  ALGEBRA. 

SURDS. 

105.  If  the  root  of  a  quantity  cannot  be  exactly 
obtained,  its  indicated  root  is  called  a  Surd  or  Irra- 
tional Quantity.  All  quantities  which  are  not  surds 
are  called  rational  quantities.  The  order  of  a  surd 
is  indicated  by  the  index  of  the  root.  Thus,  V«  and 
"ija  are  respectively  surds  of  the  second  and  ?/th 
orders.  The  surds  of  most  frequent  occurrence  are 
those  of  the  second  order;  they  are  often  called 
quadratic  surds. 

106.  Surds  of  different  orders  may  be  transformed 
into  others  of  the  same  order.  The  order  may  be 
any  common  multiple  of  the  orders  of  the  given 
surds ;  but  usually  it  is  most  convenient  to  choose 
the  L.  C.  M. 

Thus,  ^a  -(h  =  (^-  /yV, 

and  ^J^^fi  =b^  ^  ^¥. 

107.  A  surd  is  in  its  simplest  form  when  the 
smallest  possible  entire  quantity  is  under  the  radi- 
cal sign.  Surds  are  said  to  be  Like  when  they  have, 
or  can  be  so  reduced  as  to  have,  the  same  irrational 
factor ;   otherwise  they  are  said  to  be  Unlike. 

Thus,  2/y/5  and  ^\/s  ^^^  I'l^^  surds,  so  also  are  /\/i8  and  \/^. 

108.  In  adding  or  subtracting  surds  reduce  them 
to  their  simplest  form  by  the  principles  of  §  70,  and 
combine  those  that  are  like. 


SURDS.  47 

109.  The  product  or  quotient  of  surds  of  the  same 
order  may  be  obtained  by  the  laws  of  exponents 
(§  70).  If  they  are  of  different  orders  they  may  be 
reduced  to  the  same  order. 

Thus,  x^a  X  ^\/^=^^aKi 

=  xda^'^J^  =  xb  ^^*7«. 

110.  When  two  binomial  quadratic  surds  differ 
only  in  the  sign  of  a  surd  term,  they  are  said  to  be 
Conjugate. 

Thus,  ^Ja  +  ^b  is  conjugate  to  ^/a  —  ^,  or  —  ^/a  +  ^/b. 

The  product  of  two  conjugate  surds  is  evidently 
rational. 

111.  The  quotient  of  one  surd  by  another  may  be 
found  by  expressing  the  quotient  as  a  fraction,  and 
then  multiplying  both  terms  of  the  fraction  by  such 
a  factor  as  will  render  the  denominator  rational. 
This  process  is  called  rationalizing  the  denominator. 
The  cases  that  most  frequently  occur  are  the  three 
following: 

I.  When  the  denominator  is  a  monomial  surd, 
as  V^j,  the  rationalizing  factor  is  evidently  y~^ . 

II.  When  the  denominator  is  a  binomial  quad- 
ratic surd,  as  V^  +  V^,  the  rationalizing  factor  is 
its  conjugate,  ^/a  —  Vi   or  —  V^  +  V^. 


48  ALGEBRA. 

111.  When  the  denominator  is  of  the  form  "s/a 
+  "s/b  +  V^,  first  multiply  both  terms  of  the  fraction 
by  ^/a  +  ^Jb  —  "s/c;  the  denominator  thus  becomes 
(V^+ V^)2-(V?)2,  or  (^+^-^  +  2  V^.  Then 
multiply  both  terms  of  the  fraction  by  {a  ^  b  —  c) 
—  2  "s/ab ;  and  the  denominator  becomes  the  rational 
quantity  (a  -\-  b  —  cy^  —  /i^ab. 

112.  To  find  a  factor  that  will  rationalize  aiiy  given 
binomial  surd,  as  v  a  ±  'V^b. 

Let  n  be  the  L.  C.  M.  of  r  and  s ;  then  {^\/dy 
and  ('v/^)"  are  both  rational  and  so  also  is  their  sum 
or  difference.     There  are  three  cases 

I        I 

I.  When  the  given  surd  is  a''  —  b' ;    then  by  §  83 

II  l/rl  n  —X  «— 21  «— 32  «— 1 

a'  —  ^ 

=  the  rationalizing  factor. 

I  z 

II.  When  the  given  surd  is  a''  +  b%  and  n  is  even/ 
then  by  §  83 

(     ^)  '        I/O"               "— ^              >'-2    i.             "—3     1  "— ^ 

—  X  ~      V        ^  ^     '       -^^       ^^+^''      ^ ^' 

dr  +  <^'' 

=  the  rationalizing  factor. 


SURDS.  49 

j_         1 
III.    When  the  given  surd  is  a''  +  lf%  and  u  is  odi^ ; 

then  by  §  83, 

a"-  +  d' 

=  the  rationalizing  factor. 

In  each  case  the  rational  product  is  the  numerator 
of  the  fraction  in  the  first  member  of  the  identity. 

Example.    Find  the  factor  that  will  rationalize  Vs  +  V5. 

V3  +  V5  ____  ___ 

=  9V3-9\^5+3V3v^25-i5+5V3'\?^5-5v^2S. 


EXERCISE  6. 
Find  the  value  of 

1.  (\/~2  +  Vi  -  Vs)  (a/^  +  V3  +  Vs)- 

2.  (4  +  3  A/2)  -^  (5  -  3  V2)- 

3-    17  -^  (3  V7  +  2  a/3)- 

4.    (2  a/3  +  7  V2)  ^  (5  V3  —  4  A/2). 

a/3  +  \/2    .    7  +  4  V3 

2  —  V3         a/3  —  V2 

6    gV^+_^   .   8V3-6a/5 
5  +  V^    '   5V3-3V5 


50  ALGEBRA. 

Rationalize  the  denominator  of 

7.  3_+  a/6  _ 

5  V3  —  2  V12  —  V32  +  V50 


Vi  +  ^^—  a/i  —  ^'' 


Vi  +  ^"^  +  a/i  —  ^■'^ 


a/2  V^3 

12. 


V2  +  V3  -  a/s  Vz  +  A^9 

a/io  +  Vs  —  \/3  a/2  ^3 


V3  +  Vio  —  V5  A/3  +  a/2  • 

V^3  -  I  a/8  +  \/4 

II.    ^-^^^ 14.    — z ^• 

V  3  +  I  V8  —  V4 


113.  77^^  square  root  of  a  rational  quantity  cannot 
be  partly  rational  and  partly  a  quadratic  surd. 

If  possible  let     ^fa  —  ;/  +  \/m, 
then  a  ^^  71^  -\-  m  -\-  211  ^m  ; 

J—      a  —  n^  —  m 

,'.  Wm  — , 

2  71 

which    is    impossible,  since    a   surd    cannot   equal  a 
rational  quantity. 

114.  /;/  any  equality  containing  rational  quantities 
and  quadratic  surds,  the  rational  parts  in  the  two 
members  are  equal,  and  also  the  irrational  paints. 


SURDS.  51 

Suppose  a  +  ^/~b  —  x  -\-  \fy. 

If  possible  let  a  =  x  ^-  m; 

then  ^  +  w+  ^Jl^  X  ■\-  Vj/ 

or  \/y  —  m  -\-  V^, 

which  is  impossible  by  §  113. 

Hence  a  =  x^ 

and  therefore  V^  =  Vj* 

115.    To  find  the  square  root  of  ^  ±  2  Vh. 


Since       Vx  +  y  ±  2  ^Jxy  =  ^/x  ±  \fy; 


therefore  '  y  a  ±  2  \fb  —  V-^  ±  V>'»  (i) 

if  :jc  +  ^  =  «,  and  :»:  j'  —  b,  (2) 

Solving  equations  (2)  as  simultaneous  (§  166),  and 
substituting  the  results  in  (i),  we  obtain 

Equations  (2)  may  often  be  solved  by  inspection. 

Example.     Find  the  square  root  of  13  +  2  Vso* 

Here  x  -\-  y  —  13  and  xy  —  yy\ 

X  —  \o  and     y  ~  Z- 

H^"^^  V13  +  2  S'Jo  -  V"^  +  V3- 


52  ALGEBRA. 

EXERCISE   7. 

Find  by  inspection  the  square  root  of 

1.  7  —  2  Vio.         4-   i8  — SVs-  7-    19  +  8^/3- 

2.  5  +  2  V6.  5.    47-4  '^2>Z'         8.    II  +  4  V6. 

3.  8  —  2^7'  6.    15— 4Vi4-         9-    29  +  6^/22. 

IMAGINARY   QUANTITIES. 

116.  Imaginary  quantities  frequently  occur  in 
mathematical  investigations,  and  their  use  leads  to 
valuable  results.  By  the  methods  of  Trigonometry, 
any  imaginary  expression  may  readily  be  reduced 
to  the  form  of  a  quadratic  imaginary  expression. 
We  give  below  some  of  the  laws  of  combination  of 
quadratic  imaginaries. 

117,  By  the  definition  of  a  square  root  we  have 

V-^i  X  ^/—  I  =  —  I. 
.  • .     ^/a  V—  I  X  '\/a  V—  i  =  —  ^  / 
that  is,  (V^  V^y  =  —a^  {^^^af. 

By  this  principle  any  quadratic  imaginary  term 
may  evidently  be  reduced  to  the  form  c  V—  i- 


Thus,  /y/—  II  «2  —  ^i I  ^2  ^_  J  _  ^  y'l I  ^. 


IMAGINARY   QUANTITIES.  53 

118.  To  add  or  subtract  quadratic  imaginaries, 
reduce  each  imaginary  term  to  the  form  c  V—  I, 
and  then  proceed  as  in  the  case  of  other  surds. 

Thus,     V-4  +  V--9  =  2  V^.  +  3  V-^  =  5  a/-^- 

119.  To  find  the  successive  powers  of  V—  i . 

...    ^^—^Y  =.(_i)(vcr7)  =-V=T; 
...    (V^^  =(-i)(V=^)^  =  +i; 
.-.    (V=T)*''=(+ i)"  =+i, 

in  which  n  is  any  positive  integer.    Hence,  in  general, 

120.  An  expression  containing  both  real  and  im- 
aginary terms  is  called  an  Imaginary  or  Complex  ex- 
pression. The  general  typical  form  of  a  quadratic 
imaginary  expression  is  a  +  d  V—  i.  U  a  =  0,  this 
becomes  d  V—  i. 

121.  Two  imaginary  expressions  are  said  to  be 
Conjugate  when  they  differ  only  in  the  sign  of  the 
imaginary  part. 

Thus,  a  —  l>  \/—  I  is  conjugate  to  a  +  d  ^J^'x, 


54  ALGEBRA. 

122.  The  Sinn  and  product  of  two  conjugate  imagi- 
nary expressiofts  are  both  real. 

For    {a  -\-  b  ^—  \)  -\-  {a  —  b  V^^)  =  2  ^ 
and  {a-^  b  V^)  {a  -  b  V^  ■=  d"  -  {-  b"-) 

=  0"+  b'' 

The  positive  square  root  of  the  product  a'^  +  b"^  is 
called  the  Modulus  of  each  of  the  conjugate  expres- 
sions, a  ^-  b  V—  I   and  a  —  b  V—  i. 

123.  If  two  imaginary  expressions  are  equal,  the 
real  parts  must  be  equal  and  also  the  imagijiary 
parts. 

For  let  a  -\-  b  V—^  =z  c  ^  d  V^^ ; 

then  a  —  c  =  (d  —  b)  V—  i. 

Hence  (^  -  d' =  -  {^  -  b)\ 

which  is  evidently  impossible,  except  a  ^=  c  and  b  =  d. 

124.  Corollary.  U  a  -\-  b  V—  i  ==  0,  a  =  0,  and 
^  =  0. 

125.  To  multiply  or  divide  one  imaginary  expres- 
sion by  another,  reduce  them  each  to  the  typical 
form,  then  proceed  as  in  the  multiplication  or  divi- 
sion of  any  other  surds,  obtaining  the  product  or 
the  quotient  of  the  imaginary  factors  by  §  1 19. 


Thus,         /y/—  a  X  \/—  b  =  A^a  ^-  i  x  ^/b  ^—  i 

=  a/^  ^~b  ( v^)'  =  -  V^- 


IMAGINARY   QUANTITIES. 


55 


Remark.  The  student  should  carefully  note  that 
the  product  of  the  square  roots  of  two  negative  num- 
bers is  not  equal  to  the  square  root  of  their  product 

Thus,  -y/— 2  X  y'— 8  does  not  equal  '\/i6. 

126.  When  the  divisor  is  imaginary  the  quotient 
may  be  found  by  expressing  it  as  a  fraction  and  then 
rationalizing  the  denominator. 


Thus, 


I  ^  S  +  g/y/JV-i  ^i       2_V3    /— - 

3-2V"3~         9+12  7+     21      V       • 


EXERCISE  8. 
Perform  the  following  indicated  operations : 

I.   4  V—  3X2  V— 2. 

2.  (V2  +  V—  2)  (v^  —  V—  2). 

3.  (2  \/=^)^ 

4.  (2  a/-  3  +  3  V^)  (4  V^  -  S  V^)' 

5.  V— 16-^  V'-4- 

6.  (3  V^  -  5  V^  (3  V^  +  5  V^). 

7.  (i  +  V^)  ^  (I  -  V=^). 

8.    (4  +  V^^)  H-  (2  -  v'^. 

3  V^ri  —  2  V— 5  rt— V— ^ 

9-    7= 7=  •  10.    i^ j^' 

3  V—  2  +  2  V—  5  ^  +  V  — ^ 

1 1 .    What  is  the  modulus  of  3  +  2  a/— 3  ?    Of  5  —  3  V^  ? 


56  ALGEBRA. 


CHAPTER  VII. 
EQUATIONS. 

127.  An  Equation  is  an  equality  that  is  true  only 
for  certain  values,  or  sets  of  values,  of  its  unknown 
quantities.  Any  such  value,  or  set  of  values,  is 
called  a  Solution  of  the  equation.  Equations  are 
classified  according  to  the  number  of  their  unknown 
quantities;  thus,  we  have  equations  of  one  unknown 
quantity  ;  of  two  unknown  quantities ;  of  three  un- 
known quantities;   and  so  on. 

For  example,  the  equality  5  or  =  15  is  an  equation  of  one  un- 
known quantity  x ;  its  single  solution  is  x  —  2>'  Again,  {x  —  5) 
(;r  —  4)  =  0  is  an  equation  ;  its  two  solutions  are  evidently 
X  —  ^  and  X  —  \.  The  equality  j/  =  2  :r  +  3  is  an  equation  of 
two  unknown  quantities  :ir  and  y ;  one  of  its  solutions  is  ;ir  =  i, 
y  —  S\  another  is  ;ir  =  2,  /  =  7  ;  another  is  ;r  =  3,  j/  =  9  ;  and 
so  on  for  an  unhmited  number  of  solutions. 

128.  An  equation  is  said  to  be  Numerical  or  Literal. 
according  as  its  known  quantities  are  represented  by 
figures  only,  or  wholly  or  in  part  by  letters. 

129.  When  an  equation  contains  only  rational  in- 
tegral functions  of  its  unknown  quantities,  its  Degree 
is  that  of  the  term  of  highest  degree  in  the  unknown 


EQUATIONS.  57 

quantities.  Thus,  the  equations  x^  +  x^  -{-  4  =  0  and 
xy^  +  x}'  =  5   are  each  of  the  third  degree. 

A  Linear  equation  is  one  of  the  first  degree. 

A  Quadratic  equation  is  one  of  the  second  degree. 

A  Cubic  equation  is  one  of  the  third  degree. 

A  Biqiiadratic  equation  is  one  of  the  fourth  degree. 

Equations  above  the  second  degree  are  called 
Higher  Equations. 

Equivalent  Equations. 

130.  Two  equations  involving  the  same  unknown 
quantities  are  said  to  be  Equivalent  when  they  have 
the  same  solutions;  that  is,  when  the  solutions  of 
either  include  all  the  solutions  of  the  other. 

Thus,  ^x-\oa  =  ^x  —  4a  and  2x  =  6a  are  equivalent 
equations  ;  for  the  only  solution  of  either  is  jr  =  3  rt.  A  single 
equation  may  be  equivalent  to  two  or  more  other  equations. 

Thus,  (3jr-6rt)  (;r2-9^2)  =0  (l) 

is  equivalent  to  the  two  equations 

3  ;r  —  6  «  =  0,  (2) 

and  ;i-2  -  9  <J2  -  0.  (3) 

For  any  solution  of  (i)  must  evidently  render  one  of  the  factors 
of  its  first  vnember  equal  to  zero,  and  hence  must  satisfy  (2)  or 
(3) ;  and  conversely  any  solution  of  {2)  or  (3)  must  satisfy  (i). 

131.  If  the  same  quantity  be  added  to  both  members 
of  an  equation,  the  resulting  equation  will  be  equivalent 
to  the  first. 


58  ALGEBRA. 

Let  A  =  B  (i) 

represent   any   equation    of  one  or  more   unknown 

quantities,  and  let  m  denote  any  quantity  whatever ; 

then  by  §  23 

A^  7n  —  B  -^  m.  (2) 

Now  it  is  evident  that  (i)  and  (2)  are  each  satisfied 
by  any  set  of  values  of  the  unknown  quantities  that 
will  render  A  and  B  equal,  and  only  by  such  sets. 
Hence  (i)  and  (2)  are  equivalent  equations. 

132.  Corollary.  By  the  principle  of  §  131  we 
may  transpose  any  term  from  one  member  of  an 
equation  to  the  other  by  changing  its  sign.  For  this 
is  the  same  thing  as  adding  to  both  members  the 
term  to  be  transposed,  with  its  sign  changed. 

133.  If  both*members  of  an  equation  be  miiltiplied  by 
the  same  known  quantity ^  the  resulting  equation  will 
be  equivalent  to  the  first. 

Let  A^B  (i) 

represent  any  equation,  and  c  any  known  quantity; 
then  by  §  23 

cA  =  cB.  (2) 

Now  it  is  evident  that*(i)  and  (2)  are  each  satisfied 
by  any  set  of  values  of  the  unknown  quantities  that 
will  render  A  and  B  equal,  and  only  by  such  sets. 
Hence  (i)  and  (2)   are  equivalent  equations. 


EQUATIONS.  59 

134.  Corollary.    The  principle  of  §  133  is  used 

in 

(i.)  Clearing  an  equation  of  fractions  of  which 
the  denominators  are  known  quantities. 

(ii.)  Changing  the  signs  of  all  its  terms,  which  is 
equivalent  to  multiplying  both  members 
by  —  I. 

(iii.)  Dividing  both  members  by  the  same  known 
quantity,  which  is  equivalent  to  multiply- 
ing by  its  reciprocal. 

Thus,  if  we  multiply  both  members  of  the  equation 
jr  —  4  _  ;r  —  lo 
~~7  S~ 

by  35,  we  obtain  the  equivalent  equation 

5  ;r  —  20  =:  7  :r  —  70. 

135.  If  both  members  of  ajt  equation  be  multiplied  by 
the  same  integral  function  of  its  unknown  quantities^ 
in  general^  new  solutions  will  be  introduced. 

Let  A  =  B,  Qx  A-  B^^,  (i) 

represent  any  equation,  and  m  any  integral  function 
of  its  unknown  quantities ;  then  by  §  23 

mA  =  mB,  or  m  (A  —  B)  =  0.  (2) 

Now  (i)  is  satisfied  only  when  A  is  equal  to  B  ;  but 
(2)  is  satisfied  not  only  when  A  is  equal  to  B,  but  also 
in  general  when  m  =  0.  Hence  the  solutions  of  m  =  0 
have  been  introduced  by  multiplying  (i)  by  m. 


6o  ALGEBRA. 

Thus,  if  we  multiply  both  members  of  the  equation 
x  =  4,  or  X  —  4  =  0, 
by  ;ir  —  2,  we  introduce  the  solution  of  ;jr—  2  =  0;  for  we  obtain 

(x'-2)(z-4)=0, 

which  is  evidently  equivalent  to  both  ;r  —  4  =  0  and  :r  —  2  =  0. 

Again,  if  we  multiply  both  members  of  the  equation 

J/  =  2  X,  or  y  —  2x  =  0, 

hy y  —  x,  we  introduce  the  solutions  oiy—x  =  0  -,  for  we  obtain 

(/-'■^)(j  --2.r)  =  0, 
which  is  evidently  equivalent  to  both  y  —  x  =  0,  and  y  —  2x  =  0. 

136.  Corollary  i.  If  m.  is  the  denominator  of  a 
fraction  in  the  equation  A  =  B,  tJien  multiplying  both 
members  of  K  —  Vi  by  vcv  does  not,  in  general,  introduce 
new  solutions.^ 

If  no  one  of  the  solutions  o{  7n  =  0  appears  among 
those  of  the  resulting  equation,  then  evidently  no 
solution  has  been  introduced ;  if  any  of  them  do 
appear,  those  must  be  rejected  which  do  not  satisfy 
the  given  equation. 

Example.     Solve     — - —  =  e  —  x.  (i) 

X  —  L        ^ 

Multiplying  (i)  by  ^  -  i,         3  =  (-^  -  0  (S  -  ^)>     (2) 
or  x'^-6x+S  =  0. 

Hence  (x  -  4)  (x  -  2)  =  0.  (3) 

Now  the  only  solution  that  could  be  introduced  by  multiplying 
(i)  by  ;r  —  I  is  ;ir  =  I.  But  the  solutions  of  (3)  are  x=  4  and 
x=  2  '^  hence  the  solution  x=  i  was  not  introduced. 

*  The  reason  for  this  exception  to  §  135  is  that  in  this  case 
the  solutions  of  ;//  =  0  do  not  make  both  A  and  B  finite.  Thus, 
the  solution  of  .r  —  i  =  0,  or  ;r=  i,  does  not  render  the  first 
member  of  (i)  finite. 


EQUATIONS.  6 1 

To  avoid  introducing  new  solutions  in  clearing  an 
equation  of  fractions  : 

(i.)  Those  which  have  a  common   denominator 
should  be  combined. 

(ii.)  Any  factor  common  to  the  numerator  and 
denominator  of  any  fraction  should  be 
cancelled. 

(iii.)  When  multiplying  by  a  multiple  of  the  de- 
nominators always  use  the  L.  C.  M. 

Example.     Solve   i = 6.  (0 

X  —  I          I  —  X 

Transposing  and  combining,  we  have 

I  — =  -  6. 

X  —  I 

.*.     I  —  (;r  +  l)  =  -  6,  or  ;r  =  6. 

But  if  we  first  clear  (i)  of  fractions,  we  obtain 

X— I—  x'^  =  —  I— 6x+6, 

or  (x-6)(x-i)  =  0, 

of  which  the  roots  are  6  and  i.     But  as  ;r=  i  does  not  satisfy 
(i),  the  root  i  was  introduced  in  clearing  of  fractions. 

137.   Corollary  2.    In  solving  an  equation  of  the 

form 

m  A  =  m  B,  or  m  {A  —  B)  =  0, 

we  should  write  its  two  equivalent  equations, 

7Tt  =  0,  and  ^  —  ^  =  0, 

and  solve  each. 


62  ALGEBRA. 

Thus,  the  equation  ;i'3  —  i  =  0  maybe  written  in  the  form 
(x-  i)(x^  +  x+  i)  =  0,  (I) 

which  is  equivalent  to  the  two  equations 

;ir  -  I  =  0  and  x^  i-  x  +  i  =  0, 
the  solutions  of  which  are  readily  found. 

138.  If  both  members  of  an  equation  be  raised  to  the 
same  integral  pozuer,  in  general,  new  solutions  will  be 
introduced. 

Let  the  equation  be      A  =  B.  (i) 

Squaring  both  members  of  (i)  we  obtain 

^2  ^  B\  or  A^-B^  =  0, 

which  can  be  written  in  the  form 

{A  -B){A-\-B)  =  0.  (2) 

Now  (2)  is  equivalent  to  the  two  equations 

^  -  ^  =  0  and  ^  +  ^  =  0. 

Therefore  the  solutions  of  ^  +  ^  =  0  were  intro- 
duced by  squaring  both  members  of  (i). 

Hence  if  in  solving  an  equation,  we  raise  both  its 
members  to  any  power,  we  must  reject  those  solu- 
tions of  the  resulting  equation  which  do  not  satisfy 
the  given  equation. 


Example  i. 

Let  the  equatioi 

ibe 

X  — 

4- 

V4- 

-  X  — 

Squaring  (i), 

4- 

-  x  = 

x^- 

-8;r+  i6, 

or 

;r2  -  7  r  + 

12  = 

0. 

- 

Hence            ( 

x-A){x~ 

3)  = 

0. 

63 


(I) 


(2) 

Now  the  solutions  of  (2)  are  evidently  x  =  4  and  ;r  =  3,  of 
which  x=  3  does  not  satisfy  (i).  Hence  the  solution  x  =  3 
was  introduced  by  squaring  (i). 

Example  2.     Let  the  equation  be 

jy-2  =  x.  (I) 

Squaring  (i),  ( j  -  2)2  =  x'',  or  (/  -  2)2  -  x^  =  0. 
Hence  (jy  -  2  -  x)  (y  -  2  +  x)  =  0.  (2) 

Now  (2)  is  equivalent  to  the  two  equations 

J  —  2  —  r  =  0  and  ^  —  2  -}-  jr  =  0. 
Hence  the  solutions  of  j^  —  2  +  x  =  0  were  introduced  by 
squaring  (i). 

Linear  Equations  of  one  Unknown  Quantity. 

139.  Any  solution  of  an  equation  of  one  unknown 
quantity  is  called  a  Root  of  the  equation. 

140.  By  the  preceding  principles  any  Linear  equa- 
tion of  one  unknown  quantity  can  be  reduced  to  an 
equivalent  equation  of  the  form 

ax  =:  c.  ( I ) 

Dividing  both  members  of  (i)  by  ^,  we  obtain 

X  =  c-^  a. 
Hence  a  linear  equation  has  one,  and  only  one  root. 


64  ALGEBRA. 

EXERCISE  9. 

Solve  the  following  equations ;  that  is,  find  their  roots 

I. h +  ^  =  0. 

7  3  21 

4  (^  +  2)     ejx-y) 

2. =  12. 

3  7 

7  -  5:y  ^  II  -  i5>r 

I  +  ^  I  +  3:^    * 

3^—1       4J\?  —  2_i 
2:r  —  I       ^x  —  I       6 

4  (-^+3)       8jr+37       7-^—29 


5- 


18  5^—12 


30  4-  6  ^       60  4-  8  a:  48 

6.     , 1-    ; =14-1 ~ . 

^4-1  X  +  3  X  -\-  I 

Reducing  the  first  two  fractions  to  a  mixed  form,  we  have 
^         24         „         36  48 


X  +  I  ^+3  X  +  I 

36         24  32 

—  or  — ■ —  =  — ; — ,  etc. 


;r  +  3      X  +  1'    ^    X  +  -^      x+  i 

^+5       X  —  6 :r  —  4      X  —  15 

X  -\-  ^      X  —  y       X  —  5       X  —  16 

.5  ^  —  .4        2  X  —  .1 
9.    V-^  —  32  =  16  —  -y/x. 


lO. 


EQUATIONS.  65 

5^  —  9  Vs^-  3 


Vs^  +  3 


11.  ^/x  —  Vx  —  Vi  —  .T  =  I. 

In  example  10,  cancel  the  factor  common  to  the  terms  of  the 
first  fraction.  Multiplying  both  members  by  y^S^+S  would 
introduce  the  root  of  the  equation  ^5^+  ^  =  0.  In  exam- 
ple II,  neither  of  the  two  roots  obtained  will  satisfy  the  given 
equation  ;  which  therefore  has  no  root,  and  is  impossible. 

ax  —  I  \/a  X  —  I 

12.  — __ =:  4  + 

Vax  -{■  I  2 


13- 


sVx  —  4  ^  15  +  A/9-y 
2  +  ^/x         40  +  ^/x 


M-       . . F  + 


V«  —  X  •{■  \^a       \^a  —  X  —  ^a 


X 


15.    V-^— Vat  — 8 


V.^-8 


Quadratic  Equations  of  one  Unknown 
Quantity. 

141.  By  the  preceding  principles  any  quadratic 
equation  in  x  can  be  reduced  to  an  equivalent  one 
of  the  form 

ax^  ^-  bx  \-  c=^.  (1) 

In  this  equation  a  cannot  be  zero,  for  then  the 
equation  would  cease  to  be  a  quadratic. 


66  ALGEBRA. 

U  d  or  c  is  zero,  equation  (i)  assumes  the  form 
a  x^  -\-  c  =  0,  or  a  x^  +  d  X  =  0.         (2) 

Either  of  equations  (2)  is  said  to  be  Incomplete. 
The  first  is  called  a  Pure  quadratic. 

If  <^  =  ^  =  0,  (i)  becomes  ax'^  =  0 ;   .*.  ;i;  =  ±  0. 

142.    To  solve  the  pure  quadratic  a  x^  +  c  =  0. 
Solving  the  equation  for  jr^,  we  obtain 
x^  ^  —  {c  -h-  a), 
c 


±  V- 
T       a 

The  two  values  of  x  will  be  real  or  imaginary,  ac- 
cording as  c  and  a  have  unlike  or  like  signs. 

Hence  a  pure  quadratic  has  two  roots,  arithmetically 
equal  zvith  opposite  signs ;  both  are  real,  or  both  are 
imaginary. 

143.  To  solve  the  incomplete  quadratic  ax^  +  b  x  =  0. 
This  equation  may  be  put  in  the  form 

x{ax  -Y  b)=0.  .    (i) 

Now  (i)  is  equivalent  to  the  two  equations 

^r  =  0  and  a  X  +  b  =^0j 

whose  roots  are  0  and  —  (b  -^  a),  respectively. 

144.  To  solve  the  complete  quadratic 

ax^  +  bx  +  c  =  0.  (i) 


EQUATIONS.  6^ 

We  first  transform  equation  (i)  so  that  its  first 
member  shall  be  a  perfect  square.  To  do  this  we 
transpose  r,  then  multiply  both  members  by  4  a,  and 
finally  add  f^  to  both  members.     We  thus  obtain 

/^a^ x^  -\-  ^ab X  -^  b"^  ~  b'^  —  a^ac,    ■ 
or  (2  ax  +  by-  =  b^  —  ^ac. 


2ax  ■\-  b  =  ±  \b'^  —  A,ac, 


2.a 

Hence,  to  solve  a  complete  quadratic,  traitsform 
the  equation  so  that  its  first  member  shall  be  a  perfect 
square^  and  then  proceed  as  above  ;  or  put  the  equation 
in  the  form  of  {i),  and  then  apply  formula  (2). 

145.  Sum  and  Product  of  Roots.  Representing  the 
roots  of  ax'^  +  bx  +  c  =  0  by  a  and  /3,  we  have 

_  —  b  +  \lb^  —  4ac  ,  K 

—  »  \v 


2a 


Adding  (i)  and  (2),  we  find  the  sum 

-+^  =  -y  (3) 

Multiplying  (i)  by  (2),  we  find  the  product 

ap  =  '-.  (4) 


68  ALGEBRA. 

Dividing  both  members  o^  ax^  -{-  dx  -{-  c  =  0  by  {7, 
we  obtain 

^^4.*a-4-£  =  0.  (5) 

a  a 

From  (3)  and  (4)  it  follows  that  if  a  quadratic  be 
put  in  the  form  of  (5), 

(i.)    The  Sinn  of  its  roots  is  equal  to  the  coefficient  of 
X  with  its  sign  cJianged. 

(ii.)    The  product  of  its  roots  is  equal  to  tJie  known 
term. 

For  example,  the  sum  of  the  roots  of  the  equation  3:^^  +  T  x 
+  12  =  0  is  —J,  and  their  product  is  4. 

146.  Corollary  i.  If  the  roots  a  and  ^  are  arith- 
metically equal  and  opposite  in  sign,  the  equation  is 
a  pure  quadratic. 

For  \i  -b^  a^a-\-  ^  =  ^y  b^^. 

147.  Corollary  2.  If  the  roots  are  reciprocals 
of  each  other,  a  =  c ;  and  conversely,  if  a  —  c^  the 
roots  are  reciprocals.  » 

For  \{  c  ^  a-  a/3=  I,  a  =  c ;  and  conversely,  if 
a  =  c,  a  l3  =  1. 

148.  Character  of  Roots.  From  the  values  of  a  and 
0  in  (i)  and  (2)  of  §  145,  it  evidently  follows  that, 

(i.)   If  b'^  —  4ac  ^  0,    the  roots  are  real  and  un- 
equal. 


EQUATIONS.  69 

(ii.)  If  b"^  —  A^ac  =  0,  the  roots  are  real  and  equal, 

(iii.)  If  ^2  —  4^^  <  0,  the  roots  are  conjugate  im- 
aginaries. 

(iv.)  1(  d^  —  4ac  \s  a.  perfect  square,  the  roots  are 
rational;  otherwise  they  are  conjugate 
surds. 

Thus,  the  roots  of  3  jr^  —  24  :tr  +  36  =  0  are  real  and  unequal ; 

for  here 

d^  -4ac=  (;-  24)2  -  4  X  3  X  36  >  0. 

Again,  the  roots  of  3  Jir^  —  12  ^  +  135  =  0  are  conjugate  im- 
aginaries  ;  for  here 

6^-4ac=  (-12)2-4  X  3  X  135  <  0. 

149.    Number  of  Roots.      Since  -  =  —  (a  +  fi)    and 

c  ^ 

-  =  a/3,  we  may  write  the  general  equation 

a         a 

in  the  form   a:^  —  (a  +  jS)  at  +  a  y8  =  0,  (i) 

or  {x-a){x-fi)  =  ^.  (2) 

Now  a  and  /?  are  evidently  the  only  values  of  x 
that  will  satisfy  (2). 

Hence  every  quadratic  equation  has  two,  and  only 
two  roots. 

From  either  (i)  or  (2)  we  see  that  a  quadratic 
equation  may  be  formed  of  which  the  roots  shall 
be  any  two  given  quantities. 


;0  ALGEBRA. 

Thus,  if  the  roots  are  5  and  -  3,  by  (2)  the  equation  is 

(x-  s)  (x+  3)  =  0,  or  x-^  ~2x~  is  =  0. 

If  the  roots  are  2  ±  -v/— 3,  their  sum  is  4,  and  their  product 
is  7,  hence  by  (i)  the  equation  is 

x'^-  4X+  7  =  0. 

150.  Resolution  into  Factors.  Any  quadratic  ex- 
pression of  the  form  ax'^  +  dx  -{-  c  can  be  factored 
by  finding  the  roots,  a  and  /3,  of  the  equation. 

a  x'^  +  l^  X  +  c  =  0. 

For  a x^  +  If  X  +  c  =  a  Ix'^  +  ~ X  -{-  -) 

\         a  a) 

=  a{x  —  a)  {x  —  p).         §  149. 

Example.     Factor  2  x^  —  14^  +  36. 
The  roots  of  the  equation  2  x'^  —  14  r  +  36  =  0  are 
I  +  i  V^^  and  I  -  i  V-235 
hence  2x'^ -\^x-\-'>,(i^2{x-\-\  \/-23)  (:r  -  J  +  ^  V-^iO 

151.  Solution  by  a  Formula.  Any  complete  quad- 
ratic may  be  reduced  to  the  form 

x^+J>x  +  ^  =  0.  (i) 

Solving  (i),  we  obtain 

f.  (2) 


2        V    4 


EQUATIONS.  71 

Formula  (2)  affords  the  following  simple  rule  for 
writing  out  the  two  roots  of  a  quadratic  equation  in 
the  form  of  (i): 

The  roots  equal  one  half  the  coefficient  of  x  with  its 
sign  changed,  increased  and  diminished  by  the  square 
root  of  the  square  of  one  half  the  coefficient  of  x  dimin- 
ished by  the  known  term. 

Thus,  to  solve  ;r2  +  3  :ir  +  1 1  =  0,  we  have 

152.  Solution  by  Factoring.  In  solving  equations 
the  student  should  always  utilize  the  principles  of 
factoring  and  equivalent  equations. 

Example.    Solve  x^  =  \,  (i) 

Transposing  and  factoring,  (i)  may  be  written  in  the  form 

(:r-i)(:r2  +  :r+i)(r+  i)  (y-*  -  ;r  +  i)  =  0.         (2) 

The  roots  of  (2)  are  i,  —  i,  and  those  of  the  two  equations 

:r*  +  :r  +  I  =  0  and  :r2  -  :r  +  I  =  0, 

which  are  readily  solved. 

EXERCISE    10. 

By  §  145  determine  the  sum  and  the  product  of  the  roots 
of  each  of  the  four  following  equations.  By  §  148  find  the 
character  of  the  roots  of  each.     Then  solve  each  by  §  151. 

1.  5^:2  —  6:^  —  8  =  0.  3.    2x  —  x^^^, 

2.  ^2  4-  II  =  7  a:.  4.    5  jc'^  =  17:^;  —  10. 


72  ALGEBRA. 

Form  the  equations  whose  roots  are 

5.  3,  -8. 

m         n 

6.  §,  f  ^*    n'~  m' 

7-   3  ±  Vs-  ^  +  ^     ^  _ 

10 


o         ,/^  a  —  b    a  -^  b 

8.    2  ±  V—  3- 

Solve  the  following  equations  : 
II.   ^  = -.  14- 


2;c— 7       ^  —  3  7-^  —  5       2:^—13 

:r  +  4       ^V" —  2        ,  c  /I  ^ 

^2.  r— T  +  r-— :  =  6J.       15. 


^  —  4       X  —  3  Jt:  —  2        a:       ^  +  6 

-2:^—  I                    6  ^      X  -\-  a        X  —  2  a 

13. ; =   I .        16. ^~-^- , =1. 

/^x  -\-  1              X  ■\-  "]  X  —  2  a        X  -\-  a 

3                       T  I 


17. 


2(^'^-i)       4(^+0       8 


„      JC+I  ^—  I  2X—  I 

18.  — [-  ^ 

X  +   2  X  —  2  X  —   I 

'YAx  -\-  2       4  —  's/x 

19.     z^    =  = • 

4  +  ^x  ^Jx 

a+  2b  a"  4  ^^ 

20.    = -^ 

a  —  2  b       {a  —  2  b)  X        X 

21.  a^  x"^  —  2  a^  X -\- a"^  —  1  ==0, 

22.  ^  a^  X  =  {a^  —  b"^  -\-  xy. 


EQUATIONS.  73 

23.    I  —  4  Vx  —  a/?-^  +  2  =  0.  (i) 


Transposing  yji  x  ■\-  7.  and  squaring,  we  obtain 

I  -  8  y'^  +  16  r  =  7  ;r  +  2, 
or  9  jr  -  I  =  8  V-^.  (2) 

Squaring  (2),  81  :ir'^  -  18  jr  +  i  =  64  jr. 

.-.     8i;ir2-82;r+  I  =0. 
.-.     (r- I)  (81  ^-0  =  0.  (3) 

The  roots  of  (3)  are  evidently  i  and  ^.  Hence,  if  (i)  has 
any  root,  it  must  be  i  or  ^y.  But  neither  of  these  roots  satisfies 
(i)  ;  hence  (i)  has  no  root,  or  is  impossible. 

It  should  be  noted,  however,  that  if  we  use  both  the  positive 
and  negative  values  of  ^x  and  ^Jl  x  -^  2,  we  obtain  in  addition 
to  (i)  the  three  equations 


I  -  4  /y/;ir  +  V7  •**  +  2  =  0,  (4) 


+  4  V-^  -  V7  ^  +  2  =  0,  (5) 


I  +  4  y:r  +  y;  ;r  +  2  =  0.  (6) 

Multiplying  together  the  first  members  of  (i),  (4),  (5),  and 
(6),  we  obtain  the  first  member  of  (3)  ;  hence  equation  (3)  is 
equivalent  to  the  four  equations  (i),  (4),  (5),  (6).  Now  i  is  a 
root  of  (4),  and  ^\  is  a  root  of  (5),  but  neither  is  a  root  of  (6)  ; 
hence  (6)  is  impossible.  Equation  (3)  could  be  obtained  from 
(4),  (5),  or  (6)  in  the  same  way  it  was  from  (i). 


24.  2  V  4  +  \/2  ;«•*  +  JC*  =  ^  4-  4. 

25.  X  \/6  +  at"''  =  I  +  x^. 


74  ALGEBRA. 


26.   ^  -  V-^  +  I  ^  _5_ 


27.  + 


X  +    V2  —  JV'-^         -^  ~  \/2  —  X^  2 

28.    JC  +  Vi  +  ^^  = 


Vl   + 


5  (3:^—  i)        2  /- 

I  +  5  V  ^      yx 


^  —  82(jc  +  8)_3jc+  10 


3,.  _i^  +   4   _  3^ 


5  —  ;<?       4  —  ^       X  ■\-  2 

^.    ^  +  3       -^-3       ^ 
32. ; —  =  a. 

X  —  Z      ^  +  3 


33.    m  x^  —  • —  A*  =  I, 


34. 


x^  X       m^  —  AfO^ 


ini  —  2  a       2       4  a  —  6 ;« 


I  III 

'^'^     a  +  I?  +  X       a       I?       X 


36.    — — ^ —  (\/a  —  ^/b)  X  = 


37.    If  the  equation  x^— i^  — m(2x~S)  =  0  has  equal 
roots,  find  the  value  of  m. 


EQUATIONS.  •  75 

38.  Prove  that  the  roots  of  the  following  equations  are 
real: 

(i)     x"^  —  2  ax  -{■  a^  —  b"^  —  c""  =  0. 

(2)     {a  -  b  +  c)  x^  -\-  ^  {a  —  b)  X  +  {a  —  b  —  c)  =  0. 

39.  For  what  values  of ;«  will  the  equation 

;<;^  —  2  a:  (i  +  3  w)  +  7  (3  +  2  w)  =  0 
have  equal  roots? 

40.  Prove  that  the  roots  of  the  equation 

{a  -\-  c—  b)x'^  -^  2CX  -^  (b  ■\-  c  —  a)  =  (i 
are  rational. 

4T.   For  what  value  of  w  will  the  equation 

x'^  ^  bx  _m  —  I 
ax  —  c       m  +  I 

have  roots  arithmetically  equal,  but  opposite  in  sign? 

Solve  the  following  equations  : 

42.  x^  =  I.  45.   X*  -\-  I  =0. 

43.  x^  +  I  =  0.  46.   jc^  —  I  =  0. 

44.  jc^  —  I  =  0.  47.   ^^  +  I  =  0. 

48.  .r'*  +  ^^  —  4  ^  —  4  =  0. 

49.  (x^  —  8x+2)  (X^  -{-   2X  +  j)  =0, 


76  •  ALGEBRA. 

Resolve  into  factors  the  following  trinomials  : 

50.  4x^—  isx  +  s.  52.    7-^^  +  15-^  +  13- 

51.  ^x^—iix+iS.  S3'   3-^^+12:^+15. 

153.  Higher  Equations  Solved  as  Quadratics.  Higher 
equations  may  frequently  be  solved  as  quadratics  by 
a  judicious  grouping  of  the  terms  containing  the  un- 
known quantity,  so  that  one  group  shall  be  the  square 
of  the  other. 

Example.     Solve  Jir*  —  6  x^  +  5  Jt:^  +  12  ^  =  60.     (i) 
Adding  and  subtracting  4X-,  we  may  write  (i)  in  the  form 

(x^-6x^  +  gx^)  -  (4x^-  I2x)  -60  =  0, 
or  (^^  —  3  ^y^  —  4(x^  —  sx)  —  60  =  0. 

.'.     x'^-  3x  =  2  ±  8.  (2) 

The  solution  is  now  reduced  to  the  solution  of  the  two  quad- 
ratic equations  given  in  (2). 

EXERCISE  II. 

Solve  the  following  equations  : 

1.  X*  -\-  2  x^  —  ;^x^  —  4  jc  +  4  =  0. 

2.  ^'*  —  8  Ji;^  +  29  jc^  —  52  ^  +  36  =:  1 26. 

3.  ^^  —  6  ^^  +  1 1  ::>;=:  6. 

Muln'ply  both  members  by  x,  thus  introducing  the  root  zero; 
but  this  must  not  be  included  among  the  roots  of  the  given 
equation. 


EQUATIONS.  77 

4.  X*  —  2  ^^  +  X  =  380. 

5.  X*  —  4  x^  +  S  x'^  —  8  X  =  21. 

6.  4x*+^x  =  4x^  +  :^:^. 


7.   :v  +  16  —  7  V^  +  16=10  —  4  \x+  16. 


8.    2X^—  2X+2^/2X■^—  Tx+6  =  sx  —  6. 


9.   ;c2  —  ^  4-  5  \/2  -^^  -  5  -^  +  6  =  2  (3  -^  +  33)- 

10.  x^  +  4x^=  12.  J  y 

14.   :v"  +  6  =  5  X". 

11.  a:*  =  81. 

I  I 

I  .    ,    I       ^  15.    3  ^'  -  ;c^  —  2  =  0. 

13.    6x^  =  7x^  -2X~K       16.    i+8a*  +  9  V^«  =  0. 


17.    3  :«:^  —  7  +  3  V3^^^-i6T+2i  =  16  X. 


18.    8  +  9  a/(3-^  -  i)  (:c  -  2)  =  3  :c2  -  7 


9.    ;<;2  +  3  —  ^2  ^^  —  3  :t  +  2  =  I  (;c  +  i). 

a:^  \         ^/  9 

21.    a/^  +  12  +  V^  +  12  =  6. 


20 


yS  ALGEBRA- 


CHAPTER  VIII. 

SYSTEMS   OF   EQUATIONS. 

154.  A  single  equation  involving  two  or  more  un- 
known quantities  admits  of  an  infinite  number  of 
solutions. 

Thus,  of  _y  =  2  ;r  +  3,  one  solution  is  x  —  \^y  —  ^\  another 
is  jjT  =  2,  jj/  =r  7  ;  another  is  ;ir  =  3,  j  =  9  ;  and  so  on.  In  fact, 
whatever  value  is  given  to  x^  y  has  a  corresponding  value. 

Oi y  —  \x—  I,  one  solution  is  :r  =  i,  j  =  3  ;  another  is  ;ir=2, 
/  =  7  ;  another  is  x  —%  y  —  11;  and  so  on. 

Both  equations  have  the  solution  ;r  =  2,  y  =  y,  which  is 
therefore  a  solution  of  the  two  equations. 

155.  Equations  which  are  to  be  satisfied  by  the 
same  set  or  sets  of  values  of  their  unknown  quantities 
are  said  to  be  Simultaneous. 

Simultaneous  equations  which  express  different 
relations  between  the  unknown  quantities  are  said 
to  be  Independent.  Of  two  or  more  independent 
equations,  no  one  can  be  obtained  from  one  or 
more  of  the  others. 

Thus,  of  the  simultaneous  equations  (i),  (2),  and  (3),  any 
two  are  independent,  since  no  one  can  be  obtained  from  an- 
other. But  the  three  are  not  independent,  for  any  one  of  them 
can  readily  be  obtained  from  the  other  two. 


SYSTEMS  OF  EQUATIONS. 

x-2y  +  3z  =  2, 

(0 

2X-3jy+      2=1, 

(2) 

3^-5jy  +  A2  =  3- 

(3) 

79 


Thus,  by  adding  (i)  and  (2),  we  obtain  (3)  ;  and  by  subtract- 
ing (2)  from  (3),  we  obtain  (i). 

156.  A  System  of  equations  is  a  group  of  two  or 
more  independent  simultaneous  equations. 

157.  A  Solution  of  a  system  of  equations  is  any- 
set  of  values  of  the  unknown  quantities  which  will 
satisfy  each  of  the  equations. 

158.  Two  systems  of  equations  are  Equivalent  when 
they  have  the  same  solutions;  that  is,  when  every 
solution  of  either  system  is  a  solution  of  the  other. 

159.  If  each  equation  of  a  system  contains  only 
one  unknown  quantity,  the  system  is  solved  by 
previous  methods.  But  if  each  equation  contains 
two  or  more  unknown  quantities,  we  must  combine 
the  equations  of  the  system  so  as  to  obtain  finally 
an  equivalent  system  in  which  each  equation  con- 
tains only  one  unknown  quantity.  This  process  is 
called  Elimination,  and  is  dependent  on  the  following 
principles. 

160.  T/ie  equation  obtained  by  addi^ig  or  subtracting' 
any  two  equations  of  a  system  may  be  substituted  for 
either  one  of  I  hem.  • 


8o  ALGEBRA. 


Let  A  =  A^  and  B  =  B'  be  any  two  equations 
in  X,  f,  Zy  . . . ;  then  the  systems  {a)  and  {h)  are 
equivalent. 


vy  '  A±B  =  A'  ±B'j^'^ 


A  =  A'],_^  A  =  A' 

B  =  B' 


For  it  is  evident  that  systems  (^)  and  (^)  are  each 
satisfied  by  any  set  of  values  of  x,  y,  z,  .. .,  that  will 
render  A  equal  to  A^  and  B  to  B\  and  neither  is  sat- 
isfied by  any  other  set.  Hence  systems  {a)  and  {U) 
are  equivalent. 

Either  of  the  equations  of  the  given  system  may 
evidently  be  multiplied  by  any  known  quantity  be- 
fore they  are  added  or  subtracted. 

161.  Elimination  by  Addition  or  Subtraction.  This 
method  is  based  on  the  principle  of  §  i6o.  We  will 
illustrate  it  by  two  examples. 

Example  i.     Solve 

Multiplying  (i)  by  7, 
Multiplying  (2)  by  8, 
Adding  (3)  and  (4), 
or 

Similarly,  we  obtain 

By  §  160,  (5)  and  (6)  may  be, substituted  for  (i)  and  (2), 
respectively;  hence  the  solution  of  system  {ii)  is  given  in  {b). 


Zx^    8j=    25, 

(2)  J 

I2;ir—    7  J  =    22. 

2i;r+56_>/=  175. 

(3) 

Oi6x  —  s6y  =  176. 

(4) 

li7;ir=35T, 

y  =  2. 

(6)    J 

SYSTEMS    OF  EQUATIONS.  8 1 


m 
Example  2.    Solve  _  + 


^-.    (0  1 


nV      n' 

—  +  -  =  c',  (2) 

ntn'     nn'           ,  ^  ^ 

Multiplying  (i)  by  «',      -j-  +— -  =  en'.  (3) 

m'  n     nn' 

Multiplying  (2)  by  «,       -—  +  — -  =  ^  «.  (4) 


K^) 


Subtracting  (4)  from  (3),  {in  n'  -  in'  n)-  =  e  n'  -  d  n^ 

m  n'  —  m'n    ^ 

m'n-mn'     \    ^  ^ 
Similarly,  we  obtain  y  -  ^^^,_^,^  -J 

By  §  160  the  solution  of  system  {a)  is  given  in  {b). 

162.  If  (a)  be  any  system  of  equatiojis  in  which  A 
does  not  contain  x,  afid  (b)  ^  system  obtained  from 
system  (a)  /^/  substituting  A  /<?r  x  /;/  equations  (2) 
^«^  (3),  //^^//   the  systems  (a)  <?;^^/  (b)  are  eqicivaieftt. 

B  =  B'.     (2)  j^  (a)  B,  =  B^.     (2O  I  {b) 

Any  solution  of  system  (^d)  will  evidently  satisfy 
(2)  and  (3)  after  ^  has  been  substituted  for  its  equal, 
X ;  hence  any  solution  of  {a)  is  a  solution  of  {b). 
Again,  any  solution  of  {b')  will  evidently  satisfy  (2') 
and  (3')  after  x  has  been  substituted  for  its  equal,  A  ; 
hence  any  solution  of  (/^)  is  a  solution  of  {ci).  There- 
fore systems  {ix)  and  {U)  are  equivalent. 


82  ALGEBRA. 

163.  The  method  of  Elimination  by  Substitution  is 
based  on  the  principle  of  §  162.  We  will  illustrate 
the  method  by  a  single  example. 

Example.     Solve     3^'.+ 27  +  4^  =  19,  (i)^ 

2X+  sy  +  3^  =  2i,  (2)  y(a) 

3x-y  +  z  =  ^,  (3)  J 

From  (3),  y  =  2,x'  +  z-^.    (4) 

Substituting  in  (i)  and  (2)  the  value  oiy  in  (4),  we  obtain 

3X+2{2X  +  Z-^)  +  ^S=l(), 

and  2;f  +  5  (3;ir  +  £r-4)  +  35'  =  21  ; 

or  9^  + 62:  =  27,  (5)-] 

and  17X+  Sz  =  41.  (6)j 

From  (5),-  ^  =  ^1^,        (^^j 

From (6)  and  (7),   459   ^^°^^  ^  g^  =^  41.  (8)1 

From  (8),  2-  =  3.  (9)  1 

From  (7)  and  (9),  x=  \.  (10)  J 

From  (4),  (9),  and  (10),  J  =  2.  (i  i) 

By  §  162  the  systems  {b),  (c),  and  (d)  are  equivalent.  But  (d) 
with  (4)  forms  a  system  equivalent  to  (a) ;  hence  (d)  with  (4), 
or  (d)  with  (i  i),  forms  a  system  equivalent  to  (a).  Hence  the 
solution  of  system  (a)  is  x=  i,jy  =  2,  z  =  3. 


164.  If  in  the  method  of  elimination  by  substitu- 
tion, each  of  the  equations  is  solved  for  the  same  un- 
known quantity  before  the  substitutions  are  made, 
the  method  is  called  Elimination  by  Comparison. 


SYSTEMS   OF  EQUATIONS.  83 

165.  The  following  modification  of  the  method  of 
ehmination  by  addition  is  called  Elimination  by  Unde- 
termined Multipliers. 

Example.     Solve  ax^by^c,         ( i )  ^ 

G!x-^b'y=-c!.        (2)  J 

Multiplying  (2)  by  w,  and  adding  the  resulting  equation  to 
(i),  we  obtain, 

{a^md)x^{b\  m  V)y  =  c  +  (f  m.  (3) 

To  find  X,  let  m  be  determined  by  the  equation 

^  +  w  <^'  =  0 ; 


hence                                                   in  =  -b-^  ff. 

Substituting  this  value  of  m  in  (3),  we  obtain 

/        a'b\                db 

(4) 

Vc-bd 
hence                                                  *=^a'_va- 

(5) 

By  introducing  into  equation  (3)  the  condition 
a  ■\-  7n  a'  —  ^^  or  ?n  =  —  a  -h-  a\ 

we  obtain  y  =  a'b-ab''        (^> 

By  §  160,  (5)  and  (6)  may  be  substituted  for  (i)  and  (2), 
respectively ;  that  is,  the  solution  of  {a)  is  given  in  (5)  and  (6). 
Since  (i)  and  (2)  are  general  equations  of  the  first  degree  be- 
tween X  and/,  (5)  and  (6)  may  be  used  as  general  formulas  for 
solving  any  system  of  simple  equations  in  x  and  j'. 

Hence,  a  system  of  two  linear  equations  in  x  and  y  has  in 
general  one,  and  only  one,  solution. 


3- 

m 

+  ^ 

=  I, 

X 

m 

=  I. 

4- 

5 

X 

+  '-  = 

y 

30, 

9 

X 

_5  _ 
y 

■■  2. 

84  ALGEBRA. 

EXERCISE  12. 
Solve  the  following  systems  of  linear  equations  : 

1.  6x  +    4y=  236, 
SX+  i5y  =  S73' 

2.  ax  +  l^y  =  a\ 
bx  +  ay  =  b"^. 


5.    (^  _  ^)^  +  (^  +  i,)y  =  2  (a'-  b'), 
ax  —  by  =  a^  +  b^. 


6. 1 =:m  +  n,  9.        -  H ^-  -  =  36, 

^   1   ^  2   >     2 

X      y 


7.  2x+:^y  +  42  =  2g, 
3:^  +  2  V  +  52:  =  32, 
4-^  +  3  J'  +  22:=  25. 

8.  «JJ  -\-  b  X  =:  Cf 

ex  +  a  z  =  bf 
b  z  ■\-  cy  =  a. 


I 

I 

z 

28, 

3y 

I 

2  Z 

20. 

10. 

32 

■]-^U 

= 

33  J 

7^ 

—  2  z 

+  3^^ 

=:' 

17. 

4J 

—  2  z 

+     z^ 

= 

II, 

4y 

-ZU 

4-  2  z^ 

= 

9, 

sy 

-3^ 

—  2U 

= 

8. 

SYSTEMS    OF  EQUATIONS.  85 


Systems  of  Quadratic  Equations. 

166.  A  system  consisting  of  one  simple  and  one 
quadratic  equation  has  ifi  general  two,  and  only  twOy 
solutions. 

This  theorem  will  become  evident  from  the  solu- 
tion of  the  following  example: 

Example.     Solve  8;r-4j  =  — 12,        (i)^ 

3:1:2  +  2^-^  =  48.  (2)]' 

Solving  (i)  for  J,  j/  =  2  ^  +  3.    (3) 


From  (2)  and  (3),  :r2  +  2  ;i-  =  3.  (4) 

From  (4),  x=  i,  or  -  3.  (5) 

From  (3)  and  (5),  J  =  5»  or  - 3.  (6) 

From  §  162  the   systems  (rt),   {b)^  and  (^)  are  equivalent; 
hence  the  two  solutions  of  {a)  are  given  in  (<:). 


167.    A  system   of  two  complete  quadratic  equa- 
tions m  X  and  y  has  in  general  four  solutions. 

Any  such  system  can  evidently  be  reduced  to  the  form 
x^  -\-bxy-\-  cjy^  +  d x  +  ey  +/=  0,     (i)  1 
;r2  +  b'xy  +  c' y'^  +  d' x  +  €>  y  +/'  =  0.     (2)} 

Subtracting  (2)  from  (i),  and  solving  the  resulting  equation 
for  Xj  we  obtain 

(t^'  ~d)y  +  ti'  -d  '        ^^^ 

Substituting  in  (i)  the  value  of  x  in  (3),  we  shall  evidently 
obtain   a  general  equation  of   the  fourth   degree  in  y.     This 


S6  ALGEBRA. 

equation  will  give  four,  and  only  four,  values  for  y  (see  exam- 
ple of  §  153).  For  each  value  of  j  in  (3),  x  has  one,  and 
only  one,  value.  Hence  a  system  of  two  complete  quadratics 
has  in  general yi7//r  solutions.  If,  however,  c  =  c'  and  l^  =  b' ,  (3) 
will  be  a  linear  equation  in  x  and  j,  and  therefore  system  {a) 
will  be  equivalent  to  one  consisting  of  a  linear  and  a  quadratic 
equation  ;  hence  in  this  case  the  system  has  but  two  solutions. 

168.  By  §  167  the  solution  of  a  system  of  two 
complete  quadratics  involves  in  general  the  solution 
of  a  complete  biquadratic,  of  which  we  have  not  yet 
obtained  the  general  solution.  But  many  systems  of 
incomplete  quadratics  can  be  solved  by  the  methods 
of  quadratics.  We  shall  next  consider  some  of  the 
most  useful  methods  of  solving  systems  involving 
incomplete  equations  of  the  second  and  higher 
degrees. 

169.  When  the  system  is  of  the  form 

X  ±  y  —  a^  x^  ^  y'^=^a\     ^^     x^  \  y"^  =  a\ 

xy  =  bl  xy  =  b]  x+y  =  b\ 

it  may  be  solved  symmetrically  by  finding  the  values 
o(  X  +  f  and  X  —  f. 

Example  i.     Solve  x—y 


4,    (01., 

>o.     (2)  J 


xy  =:    60. 
From(i),  x'^-2xy+y''=    16.     (3) 

Multiplying  (2)  by  4,  4  xy  =  240.     (4) 

Adding  (3)  and  (4),         x^  +  2xy  +y^=  256, 
or  ^+y  =  ±  16.  (5) 


SYSTEMS    OF  EQUATIONS.  87 

Now  (i)  and  (5)  are  equivalent  to  the  two  systems 


^-J=    4,  1  x-y  =  4,       1 


x+j/^iS;]  x+j/  =  —i6.j 

Whence  x=io,]  ;r  =  —    6, 1 

y=    6;J  >/  =  -io.J 

The  values  in  (c)  evidently  satisfy  system  (a).     Equations 
(2)  and  (5)  would  form  a  system  having  other  solutions  than 
those  of  system  (a),  introduced  by  squaring  (i). 
Example  2.     Solve  x^-\-y^  =  6s,        (i) 


xy  =  28.         (2) 
Multiplying  (2)  by  2,  then  adding  ^nd  subtracting,  we  have 
x'^+  2Xf  4-j2=  121, 
and  x^-2x_y+y^=      9; 

and  ■^— /  =  ±    3- 

Now  (d)  is  equivalent  to  the  four  systems 
x-\-jy=ii,  ]x+_y=ii,      ,       _  -  i         .  -  ,    ,  . 

x-y: 


} 

=  11,  ^  x+_y=ii,     "1    x-\-jr=-u,')    ;r+J=-ll,l 

snce 

=  7,1         ;r  =  4,     1         x  =  - 4,  ]^        x  =  -7,^ 

=4;  J      j  =  7;  J      y  =  -7;j      y  =  -A' } 


Whence 

X 


y 

By  §  160  systems  {a)  and  {b)  are  equivalent,  so  also  are  sys- 
tems {c)  and  (^). 

Any  system  of  equations  of  the  form 
x'^  ±p  xy  +j2  =  ^2 
:r  ±  jK  =  ^, 

in  which  /  is  any  numerical  quantity,  can  evidently  be  reduced 
to  the  first  form  given  above. 


88  ALGEBRA. 

170.    Systems  in  which  all  the  unknown  terms  are 
of  the  second  degree  may  be  solved  as  below: 

Example.     Solve  x^  -^  xy  -\-  2j2  =  ^,  (i)  | 

2x^  —  xy  +y^  =  i6.  (2)  j  ^^^ 

Multiply  (i)  by  4,  4x^  +  ^xy  +  8y^  =  176.  (3) 

Multiply  (2)  by  11,        22  x^  —  11  xy  -j-  iiy^  =  1 76.  (4) 
Subtract  (3)  from  (4),    i8x^  -isxy  -]-  3^2  =  0. 

Factor,  {y  -  sx)  (y  -  2x)  =  0.  (5) 

Equations  (i)  and  (5)  form  a  system  equivalent  to  system  (a) 
or  to  the  two  systems  (d)  and  (c), 

x^  +  xy  +  2/2  =  44,    I  x^  +  xy  +  2y^  =  44,    } 

y-3x=0;     P  ^  y-2x  =  0',     )  ^^^ 

which  are  readily  solved.     Their  solutions  are 

x=  V2,         —  V2,  2,        -2. 

^  =  3^2,        -3^,        4,        -4- 

The  above  method  is  sometimes  applicable  to  other 
systems  than  those  of  the  class  considered  above. 


^6y.     i2)\^' 


Example.     Solve  x"^  -  2y^  =  4/, 

2,x^  +  xy  —  2_y2 
Multiply  (i)  by  4,  4:r2  -  8j2  =  i6j/.     (3) 

Subtract  (2)  from  (3),  x'^  —  xy  —  6y^  =  0. 

Factor,  (x  +  2y)  (x  -  3 j)  =  0.         (4) 

Equations  (4)  and  (i)  form  a  system  that  is  equivalent  to  (a) 
or  to  the  two  systems  (d)  and  (c), 


x^-2y^  =  4y,     I  ^2  _  2^2  ^4^,     I 


which  are  readily  solved.     Their  solutions  are 

^=0,  -4,  0,  -v^. 

.y  =  o,  2,  0,  f 


SYSTEMS    OF   EQUATIONS.  89 

EXERCISE  13. 

Solve  the  following  systems  of  equations  : 

19. 


I.   x+y=    51, 

5- 

^->'  =  3» 

xy=s^^' 

;i:2_3^_j;+/z3  — 

2.   x—y=      18, 

6. 

:^^-^J+/  =  76, 

xy=  1075. 

X  +y=  i^. 

3.     ^-y=    4, 

7- 

x^+xy+y'  =  67, 

x'  +  y'=  106. 

:r2— .r>'+/  =  39• 

4.    ^'+y=i78, 

8. 

^^— 2JCJ— j^  =  I, 

x-^y=    16. 

X  +y=  2. 

9.   3^^*^— 2/ +  5^— 2j=  28, 
x+y+  4=    0. 


10.     :r2— 3^>'+y 


30.2— a:>'+3y  =:  13.    16.  ^^y^s 

y      X       2 
X  +y  =  6. 


17. 


II. 

3^' -5/ =  28, 

3^7-4/=    8. 

12. 

jr^  4-  3  jc  J  ^    54, 

xy  +  4/=  115- 

'3- 

^V+ 5  0:^  =  84, 

^+j=    8. 

14. 

x^+y^-3  =  3^y, 

2x''-6+f  =  0. 

I 

X 

I 

y 

= 

I 

■  —  J 
3 

h^ 

I 
? 

= 

5. 
9* 

18.     x'^—2xy—y'^—    31, 
^x''^-\-2xy — y^  =^  loi. 


3i^>'-3a-2-5/=45.  ^2_^^_^^y^    g^ 


90  ALGEBRA. 

171.  When  X  and  y  are  symmetrically  involved  in 
two  simultaneous  equations,  the  system  can  fre- 
quently be  solved  by  putting  x  —  v  ■\-  w  and 
y  ■=^  V  —  w. 

Example.     Solve         ;r^  +  i/'*=  82,  (i)"l 

x-y  =  2.  (2)  J 

Put  ^=:^^-^^^  (2)^ 

and  y  =  V  —  TV.  (4) 

From  (2),  (3),  and  (4),         w  =  1.  (5) 

From  (I),  (3),  (4),  and  (5), 


or  z/  =  ±  2,  or  ±  /y/—  ID.  (6) 

From  (3),  (5),  and  (6),  x=3,-i,i±  ^/^o.      ]  ,  ^ 

>{c) 

From  (4),  (5),  and  (6),  j  =  i,  _  3,  -  i  ±  y_  10.  J 

172.    I7i  general,  system  (a)  is  equivalent  to  the  two 
systems  (b)  and  (c). 

AB  =  A'B',)      ^       A  =  A',)      ,        -^5  =  0,) 


By  §  162,  system  (a)  is  equivalent  to  the  system 

AB  =  A'B,l       ^^     {A-A')B 
B 


=  A'B,l       ^^     {A-A^)B  =  0,    I 
=  B\      S  B  =  B\  S 


which  is  equivalent  to  both  (d)  and  (c). 

The  first  equation  in  system  (<^)  is  obtained  by 
dividing  the  first  equation  in  (^)  by  the  second. 

If  ^  and  B^  cannot  each  be  zero,  system  (<:)  is  im- 
possible, and  system  {d)  is  equivalent  to  system  (<^). 


SYSTEMS    OF  EQUATIONS.  91 

This  principle  is  often  useful  in  solving  systems 
involving  higher  equations. 

Example.    Solve»r»  +  ;j^y+y  =  737I>  (Ol 

x^^x_y-\-jy^=      63.  0 

Dividing  (i)  by  (2),  xHxy+y^=  117.  (3) 

Adding  (2)  to  (3),            A-2+yz=9o.  ^^^\(d) 

Subtract  (2)  from  (3),        2xy=  54.  (5)  J 
Hence                                       x  = 


=  +9. -9>+3, -3,1,  , 
=  +3, -3, +9, -9- J 


Here  A"  =  63,  and  the  system,  ^  =  0,  ^'  =  0,  is  impossible  ; 
hence  equations  (2)  and  (3)  form  a  system  equivalent  to  (a). 
Therefore  all  the  solutions  of  (a)  are  given  in  (c). 

EXERCISE  14. 

Solve  the  following  systems  of  equations : 

1.  A«  +  y  =  637,  7.  A«  -/  =  56, 

x+y=    13.  :t- +  ATj  +  y- =  28. 

2.  x^  -\-  y  =  126, 

x^  —  xy  +  _y^  =    21. 

3.  x^  4-  J^^^*^  +  J''*  =  2128, 
x^  _j_  ^^^  -j-  ^2  __      ^^ 

4.  .V  +  ^  —  V^  =       7' 

x^  -\r  y^  \  xy^  133. 

5.  a:  +  >'  =    4, 
^4  -f  /  ==  82. 

6.  f-V-^''^?,  ,2.  ^±^  +  ^:-:^=.s, 

j^'        .r       2  x  — y      X  +  y       2 

3      _  -^^  +  ^^  =  20. 


8. 

•^  J  (-^  +  ^)  =  30» 

^*+/  =  35- 

9- 

jc«-y  =  127, 

x^y  —  j^y-^  =    42. 

10. 

5^v2- 5/  =  ^+j, 

3^^'-3/  =  -^-7- 

II. 

a--*  +y  =  272, 

x-y=       2. 

92  ALGEBKA. 


CHAPTER   IX. 

INDETERMINATE   EQUATIONS  AND  SYSTEMS,  DIS- 
CUSSION   OF    PROBLEMS,  INEQUALITIES. 

173.  An  Impossible  Equation  or  System  of  Equations 
is  one  that  has  no  finite  solution.  Such  an  equation 
or  system  involves  some  absurdity. 

Two  equations   are    said  to   be   Inconsistent  when 
they  express  relations  between  the  unknown  quanti 
ties  that  cannot  coexist.     Any  system  that  contains 
inconsistent  equations  or  embraces  more  independent 
equations  than  unknown  quantities  is  impossible. 

Thus,  |:r  +  |^-5  =  ^:r  +  ^'^;jr+8is  an  impossible  equa- 
tion ;  for  attempting  to  solve  it,  we  obtain  the  absurdity  0  =  312. 

C       ax  ^-  by  —  c^  ( i ) 

Thesystem        iy^^^  ^  ^^^  ^  ^^^         (,) 

is  impossible  ;  for  attempting  to  solve  it,  we  obtain  the  absurdity 

3  =  5- 

Equations  (i)  and  (2)  are  evidently  inconsistent. 

r    ;r  +  r=    9.  (0 

The  system  -|  2  ;r  +  j  =  13,  (2) 

[x-h  2y=  16,  (3) 

is  impossible,  for  solving  (i)  and  (2),  we  obtain  x  =  4,  y  =  $; 
substituting  these  values  in  (3),  we  obtain  not  an  identity,  but 
the  absurdity,  14  =  16.  Similarly  the  solution  of  any  other  two 
of  these  three  equations  will  not  satisfy  the  third. 


INDETERMINATE   EQUATIONS   AND   SYSTEMS.      93 

Again,  if  in  the  system 

ax  —  by  —  c,  (2)  J  ^  "^ 

we  divide  equation  (i)  by  (2),  we  obtain 
ax^by-c.  (3) 

Now  (2)  and  (3)  form  a  system  equivalent  to  system  {a) ; 
hence  {a)  has  but  one  solution.  But  by  §  166  such  a  system 
as  {a)  has  in  general  two  solutions.  System  ('/)  is  called  a 
defective  system.*  Nearly  all  the  systems  in  Exercise  14  are 
defective. 

174.  An  Indeterminate  Equation  or  System  of  Equa- 
tions is  one  that  admits  of  an  infinite  number  of  solu- 
tions. Hence  a  single  equation  containing  two  or 
more  unknown  quantities  is  indeterminate  (§  154)- 
Again,  any  system  of  equations  that  contains  more 
unknown  quantities  than  mdepejidcnt  equations  is 
indeterminate. 

Thus,  the  system  ,   ,  _  . 

a'  X  +  b'y  -\-  c'z  =  0,} 
is  indeterminate ;  for  solving  the  system  for  j  and  2  we  may  give 
to  X  any  value,  and  find  the  corresponding  values  oty  and  z. 

*  An  impossible  equation  or  system  of  equations  is,  in  general,  but 
the  limiting  case  of  a  more  general  equation  or  system,  the  solutions 
of  which  in  the  limit  become  infinity. 

Thus,  the  equation  a  x  —  b  becomes  impossible  only  when  a  =  0, 
and  then  its  root  b  -^  a  becomes  ^  -^  0,  or  infinity. 

It  will  be  seen  in  §  176  that  a  system  of  linear  equations  becomes 
impossible  only  for  a  certain  relation  between  the  coefiicients  of  its 
equations,  which  renders  both  x  and  y  infinite. 

Again,  the  general  system 

ax  —  {b  ■\-  e)y  ^^  Cy   ' 
becomes  the  defective  system  {a),  only  when  e  —  Q. 


94  ALGEBRA. 

(  3  ^  —  4y  —    9 1 
The  system  <  „  f  is  indeterminate,  for  its  equa- 

tions  are  not  independent  but  equivalent. 
Again,  the  system 

2x+3y-    z=  IS,  (i) 

3x-y  +  2z=8,  (2) 

5  ^  +  2  J  +     ^  =  23,  (3) 

is  indeterminate.  No  two  of  its  equations  are  equivalent,  but 
any  one  of  them  can  be  obtained  from  the  other  two  ;  thus,  by 
adding  (i)  and  (2)  we  obtain  (3).  Hence  the  system  contains 
but  two  independent  equations. 

175.  Sometimes  it  is  required  to  find  the  positive 
integral  solutions  of  an  indeterminate  equation  or 
system.  The  following  examples  will  illustrate  the 
simplest  general  method  of  finding  such  solutions. 

(i)  Solve  7^+  I2_y  =■  220  in  positive  integers. 

Dividing  by  7,  the  smaller  coefficient,  expressing  improper 
fractions  as  mixed  numbers,  and  combining,  we  obtain 

5^  —  3 
^+^^  +  ^^=31.  (I) 

Since  x  and  y  are  integers,  31  —  :ir  —  y  is  an  integer  ;  hence 
the  fraction  in  (i),  or  any  integral  multiple  of  it,  equals  an 
integer.  Multiplying  this  fraction  by  such  a  number  as  to  make 
the  coefficient  oi  y  divisible  by  the  denominator  with  remainder 
I,  which  in  this  case  is  3,  we  have 


Hence 


IS  J'  —  9                       >'  —  2 

■ —  _  2j/  -  I  +            -an  mteger. 

y  —  2 

—- —  _  an  mteger  =/,  suppose. 

.-.    y  =  7p  +  2. 

(2) 

[)  and  (2),                  ;r=28  — 12/. 

(3) 

INDETERMINATE  EQUATIONS   AND   SYSTEMS.      95 

Since  x  and  y  are  positive  integers,  from  (2)  it  follows  that 
/  >  —  I,  and  from  (3),  that  ^  <  3 ;  hence 

i^  =  0,  I,  2.  (4) 

From  (2),  (3),  and  (4),  we  obtain  the  three  solutions 

x=  28,  16,    4; 

J  =    2,    9,  16; 

(2)  Solve  in  positive  integers  the  system 


or 


x^y-^z^    43, 

0) 

Io:ir+  5J  +  2-2r  =  229. 

(2) 

Eliminating  z,                    8  :r  +  3  j  =  143, 

IX—  2 
y  +  2x+  — —  =  47- 

(3) 

AX  —  A.                           X  —  I 

.'.    ^—- —  =  X-  \  -{ —  =  an  mteger. 

.♦.    =  an  integer  =  ^,  suppose. 

.-.    x=2>p-\-i. 

(4) 

From  (3)  and  (4),            ^  =  45  -  8/. 

(5) 

From  (I),  (4),  and  (5),    z^sp-Z- 

(6) 

From  (6),/  >  0  ;  and  from  (5),/  <  6;  hence 

P=i,  2,3,4,  5. 
Whence  x=    4,    7,  10,  13,  16; 

^  =  37,29,21,  13,    5; 

z=    2,    7,  12,  17,  22. 
Thus,  the  system  has  five  positive  integral  solutions. 

(3)  Show  that  ax  -{-  by  =  c  has  no  integral  solutions,  if  a 
and  b  have  a  common  factor  not  a  divisor  of  c. 


96  ALGEBRA. 

Let  a  =  m  d,  b  =  n  d,  c  not  containing  dj  then 
md  X  +  n  dy  —  c, 
or  mx  +  ny  =  c  -^  d.  (i) 

Now  r  -^  </is  an  irreducible  fraction  ;  while  m  x  +  ny  is  an 
integer  for  any  integral  solution  of  (i)  ;  hence  (i)  cannot  have 
an  integral  solution. 

EXERCISE  15. 

Solve  in  positive  integers 

1.  3^+29j/=i5i.  4.    i3a:+7j  =  4o8. 

2.  3^  +  87=103.  5.    23:r  +  25J^'  =  915. 

3.  J  X -}-  i2y  =^  1^2.  6.    13  :r  +  iijj/ =  414. 

7.  6  :r  +  7  J  +  42:  =  122, 
II  jc  +  8  J'  —  6  -s:  =  145. 

8.  I2:r  —  IIJJ^  +  42:=22, 

—  4X  -\-  ^y  +  z=  IJ. 

9.  20^ —  21^^=38,  II.     13  :V  +   II -S:  =  103, 

Sy  +    42=  34'  7^  —  5J=      4. 

10.    5  :r  —  I4ji/ =  II.  12.    14^—  iij==29. 

13.  A  farmer  buys  horses  at  ^iii  a  head,  cows  at  $6g, 
and  spends  ^2256;  how  many  of  each  does  he  buy? 

14.  A  drover  buys  sheep  at  $4,  pigs  at  $2,  and  oxen  at 
;^i7  ;  if  40  animals  cost  him  $301,  how  many  of  each  kind 
does  he  buy? 

15.  I  have  27  coins,  which  are  dollars,  half-dollars,  and 
dimes,  and  they  amount  to  $  9.80 ;  how  many  of  each  sort 
have  I? 


INDETERMINATE   EQUATIONS   AND    SYSTEMS.      9/ 

176.    The  Symbols  -,7:,  and  -.     The  symbol  0  de- 

notes  absolute  zero ;  that  is,  a  denoting  any  number, 
0  =  a  —  a. 

As  a  quotient,  the  symbol  0  -^  a  denotes  that 
number  which  multiplied  by  a  equals  zero ;  hence 
0  ^  ^  =  0. 

As  a  quotient,  the  symbol  a  -^  0  denotes  that  num- 
ber which  multiplied  by  zero  equals  a;  but  any  num- 
ber, however  large,  multiplied  by  zero  cannot  exceed 

(Z 

zero.     For  this  reason  the  symbol  -  represents  that 

which  transcends  all  quantity,  or  absolute  infinity. 

As  a  quotient,  the  symbol  0  -^  0  denotes  the 
number  which  multiplied  by  zero  equals  zero ;  but 
any   number   whatever    multiplied    by   zero    equals 

zero.     Hence  the    symbol  ^  represents  any  number 

whatever.  For  this  reason  ^  is  called  the  Symbol  of 
Indetermination. 

It  will  be  seen,  however,  in  §  235  that  an  expres- 
sion may  assume  this  indeterminate  form  and  still 
have  a  determinate  value. 

Example.     By  discussing  its  solution  show  that  the  system 
ax-V  by  =  c,\  ,  . 

is  (i.)  indeterminate  if  f  =  _  =  £  ;  (i) 

a       a      c 

and  (ii.)  impossible  if    —,  =  j.  not  =  -7  •  (2) 


98  ALGEBRA. 

By  Example  of  §  165  the  solution  of  system  {a)  is 

_  b' c  —  b  c'         _ad  —a'c  ,  x 

ab'  -a'  b  '  ^  ~  a  b'  -  a' b '  ^^^ 

(i.)  If  relation  (i)  exists,  then  from  (i)  we  have  at'  —  a'  b~^, 
b'  c  —  b  c'  =  {),  ac'  —  a'  c  =.0-,  hence  the  values  of  x 

and  y  in  (3)  each  assume  the  form  -,  and  the  system 

has  an  infinite  number  of  solutions. 

(ii.)  If  relation  (2)  exists,  then  ab'  —  a'  b  —  ^,  but  neither 
b'  c  ~  b  c'  nor  a  c'  —  a'  c  is  zero  ;  hence  x  and  y  are 
infinite,  and  the  system  is  impossible. 

It  is  evident  that  the  equations  in  {a)  are  equivalent  when 
(i)  is  satisfied,  and  inconsistent  when  (2)  holds  true. 


177.  Solution  of  Problems.  The  Algebraic  Solu- 
tion of  a  problem  consists  of  three  distinct  parts : 
(i)  The  Statement  in  equations;  (2)  the  Solution 
of  these  equations;  and  (3)  the  Discussion  of  this 
solution. 

To  State  a  problem  in  equations  is  to  express  by 
one  or  more  equations  the  relations  which  exist  be- 
tween its  known  and  unknown  quantities ;  that  is,  to 
translate  the  given  problem  into  algebraic  language. 

Discussion.  The  problem  given  may  impose  on 
the  unknown  quantities  certain  conditions  that  can- 
not be  expressed  by  equations.  In  such  cases  the 
solution  of  the  equation  or  system  of  equations  may 
not  be  the  solution  of  the  problem. 

When  the  known  quantities  are  represented  by  let- 
ters it  may  happen  that  the  solution  of  the  equation 


DISCUSSION   OF   PROBLEMS.  99 

or  system  is  the  solution  of  the  problem  only  when 
the  values  of  the  known  quantities  lie  between  cer- 
tain limits. 

Again,  for  certain  values  of  the  known  quantities, 
the  problem  may  be  indeterminate,  that  is,  have  an 
infinite  number  of  solutions;  for  certain  other  values 
the  problem  may  be  impossible. 

To  discover  these  and  other  similar  facts  when 
they  exist,  and  to  interpret  negative  results  when 
they  occur,  is  to  discuss  the  solution. 

Negative  solutions  denote,  in  general,  the  opposite 
of  positive  ones.  If  the  problem  does  not  admit  of 
quantities  opposite  in  quality,  negative  solutions  of  the 
equation  or  system  are  not  solutions  of  the  problem. 

(i)  In  a  company  of  10  persons  a  collection  is  taken;  each 
man  gives  $6,  each  woman  $4.  The  amount  received  is  $45. 
Find  the  number  of  men  and  the  number  of  women. 

Let  X  and  y  denote  the  number  of  men  and  women,  re- 
spectively ; 

then  X  -{■  y—  10, 1 

and         -  6;r -I- 4/ =  45.  J  ''^ 

.-.     ^=2^,    /  =  7i  (2) 

Discussion.  System  (i)  translates  the  problem,  and  (2)  is 
its  only  solution ;  hence  the  problem  can  have  no  other  solution 
than  that  in  (2).  But  the  nature  of  the  problem  requires  that 
its  solution  shall  be  whole  numbers;  hence  the  problem  is 
impossible. 

(2)  A  and  B  travel  in  the  direction  P R  7i\.  the  rates  of  a  and 
b  miles  per  hour.  At  12  o'clock  A  is  at  /*  and  B  at  Q,  which  is 
c  miles  to  the  right  of  P.     Find  when  they  are  together. 


lOO  ALGEBRA. 

P_ Q R 

Let  distance  measured  to  the  right  from  P^  and  time  reck- 
oned after  12  o'clock,  be  regarded  as  positive. 

Let  X  —  the  number  of  hours  from  12  o'clock  to  the  time  of 
meeting. 

Then  ax—bx-\-c,  (i) 

Hence  x  — .  (^ 

a  —  0 

Discussion.  If  c  is  not  zero,  and  a"^  b,  x  is  positive ;  that 
is,  A  and  B  are  together  at  some  time  after  12  o'clock. 

If  t-  is  not  zero,  and  a  <  b,  x  \s  negative  ;  that  is,  A  and  B 
are  together  at  some  time  before  12.  o'clock. 

If  the  problem  were  to  find  at  what  time  after  12  o'clock 
A  and  B  are  together,  this  negative  solution  of  (i)  would 
not  be  a  solution  of  the  problem,  and  the  problem  would  be 
impossible. 

If  ^  =  0,  and  a^  b  or  a  Kb,  x=0\  that  is,  A  and  B  are 
together  at  12  o'clock,  but  not  before  or  after  that  time. 

If  c  is  not  zero,  and  a  =  b,  x  =  c  -^  Q,  or  absolute  infinity ; 
that  is,  A  and  B  are  not  together  at,  before,  or  after  12  o'clock, 
and  the  problem  is  impossible. 

0 
0' 

number  of  times  when  A  and  B  are  together,  and  the  problem 
is  indeterminate. 

The  fraction    7    assumes  the  form  -    by   rea- 

a  —  0  0 

son   of  tzvo    independent  suppositions;  namely,  ^  =  0 

and  a  —  b.     In  all  such  cases   a   fraction  is  strictly 

indeterminate. 


INEQUALITIES.  lOl 

INEQUALITIES. 

178.  The  algebraic  statement  that  one  quantity  is 
greater  or  less  than  another  is  called  an  Inequality. 
The  signs  used  are  >  and  its  reverse  < ;  the  opening 
being  toward  the  greater  quantity. 

U  a  —  d  is  positive,  a>  b ;  if  ^  —  ^  is  negative, 
a  <  b'  The  expression  a  >  b  >  c  indicates  that  b  is 
less  than  a  but  greater  than  c.  The  expression 
a^  b  indicates  that  a  is  either  equal  to  or  greater 
than  b. 

179.  The  following  principles,  used  in  transform- 
ing inequalities,  will  upon  a  little  reflection  becoftie 
evident: 

(i.)  An  inequality  will  still  hold  after  both  mem- 
bers have  been 

Increased  or  diminished  by  the  same  quantity; 
Multiplied  or   divided    by  the   same  positive 

quantity; 
Raised  to  any  odd  power,  or  to  any  power  if 
both  members  are  essentially  positive. 

(ii.)  The  sign  of  an  inequality  must  be  reversed 
after  both  members  have  been 

Multiplied  or  divided   by  the   same  negative 

quantity ; 
Raised  to  the  same  even  power,  if  both  mem- 
bers are  negative. 


I02  ALGEBRA. 

(iii.)  If  the  same  root  be  extracted  of  both  mem- 
bers of  an  inequahty,  the  sign  must  be  reversed  only 
when  negative  even  roots  are  compared. 

180.  In  estabhshing  the  relation  of  inequahty  be- 
tween two  symmetrical  expressions,  the  following 
principle  is  very  useful. 

If  a  a7id  b  are  uneqtial  and  real,  a^  +  b^  >  2  a  b. 

For  {a  -  by  >  0,  , 

or  a''-2ab-^b^>(};  (i) 

hence  d^  ^  b'^y  2  ab,  (2) 

(i)  Prove  that  the  arithmetical  mean  between  two  unequal 
quantities  is  greater  than  the  geometrical  mean. 
If  in  (2)  we  put  d^  —  x  and  ^^  —y_^  ^e  obtain 

— ,        x-Vy         . — 
X  -^y  •>  7.  ^xy,  or  — ^  >  ^^xy. 

(2)  Show  that  a^^b^>  d^b^ab\\ia-\-b>^. 
From  (i),  a^-ab  +  b'^>  ab. 
Multiplying  hy  a  +  b,  a^  +  b^  >  d^  b  +  a  b\ 

(3)  Show   that  the    fraction    ^  ^  /  4.  ^ \  ...  ^^      <   the 

greatest  and  >  the  least  of  the  fractions  ^,  ^,  •••,  ^,  all  the 
denominators  being  of  the  same  quality. 

Suppose  that  -/  is  the  least  and  ^  the  greatest  of  these 
fractions,  and  that  the  denominators  are  all  positive. 


INEQUALITIES.  IO3 

Then  ^  =  J,  .-.     ^i-^iX^i; 

T-  ^  -7    J  •   •       "-2  ^  ^2  X   V-  > 

Adding,    ^^  +  ^3  +  •  •  •  +  ^«  >(/^i  +  ^2  +  •  •  •  +  /^;,)  -^^ . 

^1  +y^2  +  •••  +  ^«      <^i' 

Similarly  we  may  prove  that 

^1  +  ^2  + H  g>t  ^  a^ 

In  like  manner  the  principle  may  be  proved  when  all  the 
denominators  are  negative. 

EXERCISE   16. 

1.  Show  that  the  sum  of  any  fraction  and  its  reciprocal  is 
numerically  greater  than  2. 

The  letters  denoting  unequal  positive  numbers,  show  that 

2.  a^  +  d'^  -\-  c^>  ad  +  ac+  dc;     m^  +  i  >  m"^  +  m. 

3.  a^  +  /^'  +^  >  K^'^  +  «^^  +  «V  +  «r^  +  ^V  +  dc^). 
a  +  d       2  ab         a         b        i        i 

4-  ~7~>^T^'  ^  +  ;r^ ^'b'^a' 

5.  If    a^  -\-  b"^  -\-  c^  =  I,    and     m^  +  n^  +  r^  =  i,    show 
that   am  -\-  b  n  +  cr  <i  i. 

6.  If  5  ^  —  6  <  3  ^  +  8,  and  2Jt:+i<3:^  —  3,  show 
that  the  values  of  x  lie  between  4  and  7. 

7.  If     3  .r  —  2  >  ^  ^  —  f ,      and      I  —  ^  x  <  S  —  2  x, 
show  that  the  values  of  x  lie  between  1 1  and  -^/. 


104  ALGEBRA. 


CHAPTER   X. 

RATIO,    PROPORTION,    VARIATION. 

181.  The  Ratio  of  one  abstract  quantity  to  another 
is  the  quotient  of  the  first  divided  by  the  second. 

When  the  quotient  a  -^  b  or  t  is  spoken  of  as  a 

ratio,  it  is  often  written  a :  b,  and  read  '*  a  is  to  h;" 
a  is  called  the  Antecedent  and  b  the  Consequent  of  the 
ratio. 

182.  By  §  48  the  value  of  a  ratio  is  not  changed 
by  multiplying  or  dividing  both  its  antecedent  and 
consequent  by  the  same  quantity. 

Two  ratios  may  be  compared  by  reducing  them  as 
fractions  to  a  common  denominator. 

183.  When  two  or  more  ratios  are  multiplied  to- 
gether they  are  said  to  be  compounded. 

Thus  the  ratio  compounded  of  the  three  ratios  2:3,  a  :  d, 
and  b  :  e'^  is  2  a  b  :  ^  de\ 

The  ratio  a^  :  ^^  compounded  of  the  two  identical 
ratios  a  :  b  and  ^  :  ^,  is  called  the  Duplicate  ratio  of 
a  :  b.  Similarly  a^  :  b^  is  called  the  Triplicate  ratio  of 
a  :  b.  Also  a"^  :  b'^  is  called  the  Subduplicate  ratio 
of  ^  :  ^. 


RATIO.  105 

184.  The  ratio  of  two  positive  quantities  is  called 
a  ratio  of  greater  or  less  inequality  according  as  the 
antecedent  is  greater  or  less  than  the  consequent. 

185.  A  ratio  of  greater  inequality  is  diminished^ 
and  a  ratio  of  less  inequality  is  increased,  by  adding 
the  same  positive  quantity  to  both  its  terms. 

Let  Uy  b,  and  x  be  any  positive  quantities; 
then  a  +  X  \  b  ^  X  <or  >a  '.  b, 

according  as  a>  or  <,b. 

aa-\-x_x(a  —  b)  ,. 

Now  the  second  member  of  (i)  is  evidently  positive 
or  negative  according  as  ^  >  or  <  ^. 

TT  ^  +  -^  ^ 

Hence  7— —  <  or  >  -,  accordmg  as  ^  >  or  <  b. 

In  like  manner  it  may  be  proved  that  a  ratio  of 
greater  inequality  is  increased,  and  a  ratio  of  less  ine- 
quality is  diminished,  by  stib trading  the  same  qna?ttity 
from  both  its  terms. 

XS6.  By§68,    |:£Hff 

Hence,  unless  surds  are  involved,  the  ratio  of  two 
fractions  can  be  expressed  as  a  ratio  of  two  integers. 

If  the  ratio  of  any  two  quantities  can  be  expressed 
exactly  by  the  ratio  of  two  integers,  the  quantities 
are  said  to  be  Commensurable ;  otherwise  they  are 
said  to  be  Incommensurable. 


I06  ALGEBRA. 

187.  Ratio   of  Concrete   Quantities.      If  A   and  B  be 

two  concrete  quantities  of  the  same  kind,  whose  nu- 
merical measures  in  terms  of  the  same  unit  are  a  and 
b,  then  the  ratio  of  ^  to  ^  is  the  ratio  of  a  to  b. 

If  A  and  B  are  incommensurable,  that  is,  cannot  be 
exactly  expressed  in  terms  of  the  same  unit,  we  can 
ahvays  find  two  integers  whose  ratio  differs  from  that 
oi  A  \.o  B  by  as  Httle  as  we  please. 

For  divide  B  into  n  equal  parts ;  let  ^  be  one  of 
these  parts,  so  that  B  —  n^.  Suppose  ^  is  con- 
tained in  A  more  than  m  times  and  less  than  m  +  i 
times ;  then  it  is  axiomatic  that 

A\  B  >  7n  (i :  n  P  diud  <  (w  +  i)P:nP; 

that  IS,  the  ratio  A  :  B  lies  between  —  and  . 

Hence  the  ratio  of  ^  to  ^  differs  from  that  of  m  to 
n  by  less  than  i  -^ ;/,  which  by  increasing  n  can  be 
made  as  small  as  we  please. 

The  ratio  oi  A  \.o  B  is  the  fixed  value  to  which  the 
ratio  of  m  to  n  approaches  indefinitely  near  when  n 
is  increased  without  limit. 

PROPORTION. 

188.  Four  quantities,  a,  by  c,  d,  are  said  to  be  pro- 
portional if  the  ratio  a  \  b  \s  equal  to  the  ratio  c  :  d. 

The  proportion  is  written 

a  :  b  =  c  :  d,     a  :  b  :  :  e  :  d,     or  —  =  - , 

0       a 

and  is  usually  read  "  a  is  to  h  as  z  is  to  d." 


PROPORTION.  107 

The  first  term  and  the  last  are  called  the  ExtrenieSy 
and  the  other  two  the  Means  of  the  proportion. 

189.  Continued  Proportion.  The  quantities  a,  by  r, 
d,   . . . ,  are  said  to  be  /;/  continued proportio7i  if 

a\  b  —  b '.  c  — c  \  d  — '"  (i) 

In  (i),  ^  is  said  to  be  a  mean  proportional  between 
a  and  r,  and  c  a  thii'd  proportional  to  a  and  b.  Also 
b  and  c  are  said  to  be  two  ineaji  proportionals  between 
a  and  d,  and  so  on. 

190.  If  four  quantities  are  in  proportion,  the  product 
of  the  extremes  is  equal  to  the  product  of  the  means. 

If   T  =  -, »  then  by  §  23  a  d  =  c  b. 
o      a 

191.  Corollary.  \i  a\  b  =  b\  c,  then  I^  =  ac, 
or  b  =  Va  c. 

192.  Conversely,  if  tRe  product  of  two  quantities 
equals  the  product  of  tivo  other  quantities,  two  of  them 
may  be  made  the  extremes  and  the  other  two  the  means 
of  a  proportion. 

\i  a  d  =  c  b,  then  by  dividing  both  members  hy  b  d 
we  obtain  a  \  b  —  c\  d. 

193.  If  four  quantities  are  in  proportion,  they  are  in 
proportion  by 


I08  ALGEBRA. 

(i.)    Inversion :  7/"  a  :  b  =  c  :  d,  the7i  b  :  a  =  d  :  c. 

(ii.)    Alternation  :  7/"  a  :  b  =  c  :  d,  then  a  :  c  =  b  :  d. 

(iii.)    Composition :  //"a  :  b  =  c  :  d,  theji    a  +  b  :  b 
=  c  +  d  :  d. 

(iv.)    Division :    If   a  :  b  =  c  :  d,     then     a  —  b  :  b 
=  c  -  d  :  d. 

(v.)    Composition  and  Divisiori:    If  a  :  b  =  c  :  d, 
then  a  +  b:a  —  b  =  c  +  d:c  —  d. 

These  propositions  and  those  of  §§  194  and  195 
are  easily  proved  by  the  properties  of  fractions  or 
by  §§  190  and   192. 

194.  7^  a  :  b  =  c  :  d,  and  e  :  f  =  g  :  h,  then  a  e  :  b  f 
=  c  g  :  d  h. 

195.  //■  a  :  b  =  c  :  d,    then 
(i.)    ma:mb  =  nc:nd; 

(ii.)    ma:nb  =  mc:nd; 
(iii.)    a"  :  b"  =  c"  :  d",  n  being  any  exponent. 

196.  If  we  have  a-  series  of  equal  ratios,  the  sum  of 
the  antecedents  is  to  the  sum  of  the  consequents  as  any 
one  antecedent  is  to  its  consequent. 

T-    ^  ace 

L«t  ^=;/=7  =  •••'"'• 

then  a  =  bry  c=  dr,  e  =fr, ... 


PROPORTION.  109 

Adding  these  equations,  we  obtain 

«  +  ^  +  ^+  •••  =  (^  +  ^  +  /+---)^. 
a  +  c  -i-  e  +  •"  _     _^_ 

Remark.  The  method  of  proof  used  above  might 
be  employed  in  §  §  193,  194,  195.  The  proof  in  the 
next  article  further  exhibits  the  directness  and  sim- 
plicity of  this  method. 

^^  a       c      ,         ma  ■\-  nb      m  c  -\-  n  d 

197.  If  7  =    ,,  then  — , r  —  — — \ 3  • 

b      d  pa  -\-  gb        pc  -\-  q d 

_       a       c  .        ma-\-nb      mbr  +  nb      mr-\-n 

Let  1  =  ~j  =  ^;  then  - — r— r  =  ,  t — ; — r  =  t — ; —  > 
b      d  pa-\-qb        bpr-\-qb     pr-\-q 

tnc  •\-  nd      mdr  -\-  n  d      m  r  ■{•  n 
pc-\-qd        pdr  +  qd        p  r -\- q 

ma  -\-  nb  _mc  -\-  nd 
pa  -\-  qb       pc  ^  qd 

198.  Proportion  of  Concrete  Quantities.  If  ^,  B^  be 
two  concrete  quantities  of  the  same  kind,  whose  ratio 
is  a  :  b,  and  C,  D,  be  two  other  concrete  quantities  of 
the  same  kind  (but  not  necessarily  of  the  same  kind 
as  A  and  B)^  whose  ratio  is  ^  :  ^;  then 

A.B^C'.D, 

when  a\  b  —  c\  d. 

The  last  proportion  can  be  transformed  according 
to  the  theory  of  proportion  given  above,  and  the  re- 
sult interpreted  with  respect  to  A,  B,   C,  D. 


and 


no  ALGEBRA. 


VARIATION. 


199.  If  the  relation  between  y  and  z  is  expressed 
by  a  single  equation,  then  y  and  z  have  an  infinite 
number  of  sets  of  values  (§  154),  and  are  called 
variables. 

There  are  an  infinite  number  of  ways  in  which  one 
variable,  y,  may  depend  upon  another,  -z.  For  exam- 
ple, we  may  have  y  =  az,  y=:az  +  c,y  =  az'^  +  bz  +  c, 
and  so  on.  In  this  chapter  we  shall  consider  only 
the  simplest  relation,  y  =  az,  in  which  z  denotes  any 
variable,  and  a  is  a  constant ;  that  is,  a  has  the  same 
value  for  all  values  of  j^  and  z. 

200.  \{  y  =^  a  z,  the  ratio  oi  y  \o  z  is  the  constant 
a.  The  expression  j/  oc  2;  denotes  that  the  ratio  of 
y  \.o  z  is  some  constant,  and  is  read  '' y  varies  as  zT 
The  symbol  oc  is  called  the  Sign  of  Variation, 

Hence,  if  y  —  a  z,  y  01  z,  and  conversely. 

1(  y  cc  z,  or  y  =  a  z,  then  any  set  of  values  of  y 
and  z  are  proportional  to  any  other  set ;  for  the  ratio 
of  each  set  is  the  constant  a. 

Hence  _^  oc  ^  is  often  read  ''y  is  proportional  to  ^." 


I  X 

201.    If  in  §  200,  z  =  - ,  XV,  -,  X  -\-  7/,  we  have 

"^  XV 

(i.)  If  Y  =  a.  ~ ,  then  y  oc  -,  and  conversely. 

y  oz  I  -^  ;r  is  often  read  '' y  varies  inversely  as  x' 


VARIATION.  1 1 1 

(ii.)  If  y  =  axv,    t/ien  y  oc  x  v,   and  conversely, 
y  oi  xv\^  often  read  '' y  varies  as  x  and  v  jointly ^ 

X  X 

(iii.)  y/"  y  =  a  -,  then  y  cc  -  ,  and  co?iversely. 

y  cc  X  -^  V  is  often  read  ''y  varies  directly  as  x  and 
inversely  as  v!' 

(iv.)  i^  y  =  a  (x  +  v),  then  y  oc  x  +  v,  and  conversely. 

202.  The  simplest  method  of  treating  variations  is 
to  convert  them  into  equatiofis.  Of  the  six  following 
propositions  we  give  the  proof  of  the  first,  and  leave 
the  proof  of  the  others  as  an  exercise  for  the  student. 

(i.)    If  \x  oc  y,  and  y  oc  x,  then  u  oc  x. 

By  §  200,  ti  —  ay,  y  =  b  x  {a  and  b  being  con- 
stants), .*.  ti  =  a  b  X,  or  n  ^  x. 

(ii.)    //■  u  X  X,  and  y  oc  x,  then   u  ±  y  oc  x,  and 
u  y  X  x^. 

(iii.)  If  n  cfi  -K,  and  z  x  y,  then  u  z  x  x  y. 

(iv.)  Ifw  X  X  y,  //^^//  X  X  u  ^  y,  and  y  x  u  -H  x. 

(v.)  If  \i  cc  -x.,  then  z  u  x  z  x. 

(vi.)  If  \i  cc  -K,  then  u"  x  x". 

203.  //■  u  X  x  w/?^;^  y  /i"  constant,  and  u  x  y  when 
x  w"  constant y  then  u  x  x  y  when  both  x  ^«rtf  y  ^r^ 


112  ALGEBRA. 

The  variation  of  u  depends  upon  that  of  both  x 
and  y.  Let  the  variations  of  x  and  y  take  place 
successively ;  and  when  x  is  changed  to  x^^  let  u  be 
changed  to  ?/ ;  then  since  u  c^  x  when  y  is  constant, 
by  §  200  we  have 

Next  let  y  be  changed  to  ^j,  and  in  consequence 

let  71  be    changed  from   u'  to  u^ ;   then,  since  u  oz  y 

when  X  is  constant, 

u^  __y 


(2) 


Multiplying  (i)  by  (2), 
hence  by  §  200  u  oc  jirjj', 


This  proposition  is  illustrated  by  the  dependence  of  the 
amount  of  work  done,  upon  the  number  of  men,  and  the  length 
of  time. 

Thus,  Work  oc  time  (number  of  men  constant). 
Work  X  number  of  men  (time  constant). 
Work  X  time  X  number  of  men  (when  both  vary). 

The  proposition  given  above  can  easily  be  extended 
to  the  case  in  which  the  variation  of  ii  depends  upon 
that  of  more  than  two  variables.  Moreover  the  varia- 
tions may  be  either  direct  or  inverse. 

Example.  The  time  of  a  railway  journey  varies  directly  as 
the  distance,  and  inversely  as  the  velocity;  the  velocity  varies 
directly  as  the  square  root  of  the  quantity  of  coal  used  per  mile. 


VARIATION.  1 1 3 

and  inversely  as  the  number  of  cars  in  the  train.  In  a  journey 
of  25  miles  in  half  an  hour,  with  18  cars,  10  cwt.  of  coal  is  re- 
quired ;  how  much  coal  will  be  consumed  in  a  journey  of  21 
miles  in  28  minutes  with  16  cars  ? 

Let  /  =  the  time  in  hours, 

d=  the  distance  in  miles, 

V  =  the  velocity  in  miles  per  hour, 

n  =  the  number  of  cars, 

^  =  the  quantity  of  coal  in  cwt. 

Then  /x  -,  and  v  ccYl. 

^  n 

nd  nd 

,'.     /oc— =,  or /  =  ^  — —  .  (i) 

Now  ^  =  10  when  d=  25,  /  =  i,  and  n  =  18  ;  hence  from  (i) 

1  18x25  y\/To  .     -_     y\/To        nd 

2  ^s/Vo  36  X  25  36  X  25     ^q 

Hence  when  ^=21,  t  —  \%  and  n—  16,  we  have 

28  _  /y/io  X  16  X  21  _ 

^~     25x36^^    '    *"'  ^~    ^' 
Hence  the  quantity  of  coal  consumed  is  6|  cwt. 


EXERCISE   17. 

1 .  \i  a  \  b  —  c  \  d^  and  b  ;x  =  d:y^  prove  that  a  :  x  =:  c.  y, 

2.  li  a  :  b  =^  b  '.  c^   prove  that  a  :  c  ^^^  a^ :  b'^  =^  b^ :  c^. 

3.  \i  a  \  b  ■=  b  '.  c  =^  c  \  d^  prove  that  a  :  d  ^=^  a^  -.  b^  z=  ... 
Let  r  =.  a  -^b  ;  then  a  =^  b  r,  b  r=  c  r,  c  —  dr. 

.'.     abc—bcdr^j    .-.  a -^  d  =  r^  —  a^ -^  b^  =  ,., 


114  ALGEBRA. 

li  a  :  b  =  c  \  d,  prove  that 
4.    a^c+  ac'':b''d+  b d'^  =  {a -\-  cf :  {b  +  df. 


5.  a  —  c.  b  —  d=  ^a^  +  r  :  \/b'  +  d\ 

6.  V^M^:  V^''  +  ^'=\/^^+~:\/bd+~. 

7.  If  «,  ^,  c,  d,  be  any  four  numbers ;  find  what  quantity 
must  be  added  to  each  to  make  them  proportional. 

8.  If  y  varies  as  x,  and  y  —  8  when  x  =  1^  ;  fin*:  y  when 
a:  =  10. 

9.  If  y  varies  inversely  as  x,  and  y  =  J  when  ^  =  3  ;  find 
^  when  jc  =  2^. 

10.  If  ti  varies  directly  as  the  square  root  of  x,  and  in- 
versely as  the  cube  of  y,  and  if  z/  ~  3  when  x  =  256  and 
y  =  2  ;  find  jjc  when  2/  =  24  and  y  =  \. 

11.  If  2/  varies  as  x  and_y  jointly,  while  x  varies  as  -s:^,  and 
y  varies  inversely  as  u  ;  show  that  ti  varies  as  z. 

12.  The  pressure  of  wind  on  a  plane  surface  varies  jointly 
as  the  area  of  the  surface,  and  the  square  of  the  wind's  veloc- 
ity. The  pressure  on  a  square  foot  is  i  lb.  when  the  wind  is 
moving  at  the  rate  of  15  miles  per  hour.  Find  the  velocity 
of  the  wind  when  the  pressure  on  a  square  yard  is  16  lbs. 


THE   PROGRESSIONS.  II5 


CHAPTER  XI. 
THE    PROGRESSIONS. 

204.  An  Arithmetical  Progression  is  a  series  of 
quantities  in  which  each,  after  the  first,  equals  the 
preceding  plus  a  common  difference. 

The  common  difference  may  be  positive  or  nega- 
tive. The  quantities  are  called  the  terms  of  the 
progression. 

205.  Let  d  denote  the  common  difference,  a  the 
first  term,  and  /  the  «th,  or  last  term. 

Then  by  definition 

the  2d  term  =  a  -\-  d, 
the  3d  term  —  a  -\-  2  d, 
and  the  «th  term  —  a  -\r  {11  —  i)  d. 

Hence  i  =  a -\- {n  —  i)  d.  (i) 

Let  ^  denote  the  sum  of  the  terms ;  then 

S  =  a  +  {a  -\-  d)  +  (a  ^  2  d)  -{-  ...  +  {I  -  d)  ^  I, 
S=  I^  {I  -d)  -^{i  -  2d)  ■\-  ...  +  {a-^  d)  -]-  a. 

Adding  these  two  equations,  we  obtain 

2S=n{a-\-  I), 


1 6  ALGEBRA. 

Hence  S  =^  {a  +  i).  (2) 

From  (i)  and  (2), 

S=~{2a  +  (fz-i)d}.  (3) 


If  any  three  of  the  five  quantities,  a,  I,  d,  n,  5,  be 
given,  the  other  two  may  be  found  by  the  formulas 
given  above. 

206.  The  m  terms  lying  between  any  two  terms  of 
an  arithmetical  progression  (A.  P.)  are  called  the  m 
Arithmetical  Means  between  the  two  terms. 

207.  To  insert  m  arithmetical  means  between  a 
and  b. 

Calling  a  the  first  term,  b  will  evidently  be  the 
{in  +  2)th  term;   hence  from  (i)  of  §  205, 

b  =  a  -\-  (m  -\-  i)d. 

J       b  —  a 
.-.     d= , 

Hence  the  required  terms  are 

,    b  —  a  2  (b  ~  a)  ^   m  (b  —  a) 

a+  — ,     a -\ ^ — ■ -,    ...,    a -^ 

m  +  I  m  -\-  1  m  +  1 

208.  Corollary.    If  ;;^=  i,  the  arithmetical  mean 

b  —  a         a  -\-  b 

is  ^  H ^- ,  or 

I  -f  I  2 


THE  PROGRESSIONS.  II7 

209.  If  any  two  terms  of  an  A.  P.  are  given,  the 
progression  can  be  entirely  determined ;  for  the  data 
furnish  two  simultaneous  equations  between  the  first 
term  and  the  common  difference. 

Example.  The  54lh  and  4th  terms  of  an  A.  P.  are  —  61 
and  64;  find  the  27th  term. 

Here  —  61  =  the  54th  term  =  /z  +  53  ^y 

and  64  =  the    4th  term  =  a  +    3  ^. 

Hence  ^=  — -|,              ^  =  71^. 

. •.  27th  term  =  a  +  26^=6^. 

210.  A  Geometrical  Progression  (G.  P.)  is  a  series  of 
quantities  in  which  each,  after  the  first,  equals  the 
preceding  multiplied  by  a  constant  factcr.  The  con- 
stant factor  is  called  the  ratio  of  the  progression. 

211.  Let  r  denote  the  ratio,  a  the  first  term,  /  the 
«th,  or  last  term ;  then 

the  2d  term  =  ^r, 
the  3d  term  =  a  r^. 
and  the  ;/th  term  =  a  r"-\ 
Hence  /=ar*'-\  (i) 

Let  5  denote  the  sum  of  the  terms ;   then 
S^za  +  ar+ar'^  +  ar^-] +  ^r""^ 


=  a(i  +  r+r^  +  ...  +  r"-i) 


Hence  3=^-^^::^^:^.  (2) 


/'—-I 


Il8  ALGEBRA. 

1(  r  <  I,  formula  (2)  is  usually  written 

S='Lii-=^.  (3) 


From  (3),        S  = 


I  —  r 

a  ar^ 


1  —  r 


Now  \i  r  <  I,  then  the  greater  the  value  of  n,  the 

a  r" 
smaller  the  value  of  f'y  and  consequently  of  . 

Hence,  if  n  be  increased  without  limit,  the  sum  of 

the   progression   will    approach   indefinitely  near  to 
a 


1  —  r 

That  is,  the  limit  of  the  sum  of  an  infinite  number 

of  terms   of  a  decreasing  G.  P.   is   ,    or   more 

briefly,  the  sum  to  infinity  is 


I  —  r 


212.  The  m  terms  lying  between  two  terms  of  a 
G.  P.  are  called  the  m  Geometrical  Means  between 
the  two  terms. 

213.  To  insert  m  geometrical  means  between  a 
and  b. 

Calling  a  the  first  term,  b  will  be  the  i^  +  2)th 

term;  hence  by  (i)  of  §211. 


=0 


1 


(i) 


Hence  the  required  terms 'are  ar,  ar'^y  "-  ar*",  in 
which  r  has  the  value  in  (i). 


THE  PROGRESSIONS.  II9 

214.  Corollary.  l(m  =  ijr=y-y,  and  there- 
fore ar=  ^/a  b ;  hence  the  geometrical  mean  between 
a  and  b  is  the  mean  proportional  between  a  atid  b. 

215.  A  series  of  quantities  is  in  Harmonic  Progres- 
sion (H.  P.)  when  their  reciprocals  are  in  A.  P. 

Thus,  the  two  series  of  quantities, 

i»i,  \^\^  •••,  and  4, -4, -|,  ..., 

are  each  in  H.  P.,  for  their  reciprocals, 

I,  3.  S»  7.  •••,  and  i  -i  -|,  ..., 

are  in  A.  P.' 

We  cannot  find  any  general  formula  for  the  sum  of 
any  number  of  terms  of  an  H.  P.  Problems  in  H.  P. 
are  generally  solved  by  inverting  the  terms  and  mak- 
ing use  of  the  properties  of  the  resulting  A.  P. 

(i)  Continue  to  3  terms  each  way  the  series  2,  3,  6. 

The  reciprocals  ^,  ^,  ^  are  in  A.  P. ;    .-.    d=—\. 

.'.    TheA.  P.  is  I,  1,1,^,^,^,0,-^,-}. 

.-.    The  H.  P.  is  I,  f,  f,  2,  3,  6,  CO  ,  -6,  -3. 

(2)   Insert  4  harmonic  means  between  2  and  12. 
The  4  arithmetical  means  between  \  and  ^-^  are  ^^tj,  },  J,  J ; 
hence  the  harmonic  means  required  are  2f,  3,  4,  6. 

216.  If  //  be  the  harmonic  mean  between  a  and  by 
then  by  §  215, 


I 
a 

T 

I 
1' 

are 

in 

A.  P. 

I 

I 

I 

I 

'  * 

~H~ 

a 

~b 

7^' 

2 

a 

I 
b' 

or  . 

i^ 

2  a 
a-\- 

b 
■b 

1 20  ALGEBRA. 

217.  Corollary.  If  A  and  G  be  respectively  the 
arithmetical  and  the  geometrical  mean  between  a  and 
by  then  (§  §  208,  214,  216) 

A  — ,     G  —  \a  b,     H  — 


a  ■\-  b 


^^      a  -{-  b        2  ab  ,        ^„ 

.-.   A  X  11=.  — —  X  — —7  =ab=  G\ 
2  a  -{•  0 

Hence       A.G=G:B; 

That  is,  the  geometrical  mean  between  two  num- 
bers is  also  the  geometrical  mean  between  the  arith- 
metical and  harmonic  means  of  the  numbers. 

EXERCISE  18. 

1.  Sum  2,  2>\)  4ij  •••>  to  20  terms. 

2.  Sum  f,  f,  i^g?  '"^  to  ^9  terms. 

3.  Sum  a  —  2ib,   2a  —  ^by   3  ^  —  7  <^,  •  •  • ,  to  40  terms. 

4.  Sum  2  a  —  b,   ^a  —  ^b,   6  a  —  ^  b,   •  •  • ,   to  «  terms, 

5.  Insert  17  arithmetical  means  between  3^  and  —  41^^. 

6.  The  sum  of  15  terms  of  an  A.  P.  is  600,  and  the  com- 
mon difference  is  5  ;  find  the  first  term. 

7.  How  many  terms  of  the  series  9,  12,  15,  •••,  must  be 
taken  to  make  306  ? 

8.  Sum  i,  ^,  f,  •••,  to  7  terms. 

9.  Sum  I,  VSj  3j  •••>  to  12  terms. 


THE   PROGRESSIONS.  121 

10.  Insert  5  geometrical  means  between  3f  and  40^. 

11.  Sum  to  infinity  f,  —  i,  |,  ••• 

12.  Sum  to  infinity  3,  Vs^  i,  ••• 

13.  The  5th  and  the  2d  term  of  a  G.  P.  are  respectively 
81  and  24;  find  the  series. 

14.  The  sum  of  a  G.  P.  is  728,  the  common  ratio  3,  and 
the  last  term  486  ;  find  the  first  term. 

15.  The  sum  of  a  G.  P.  is  889,  the  first  term  7,  and  the 
last  term  448  ;  find  the  common  ratio. 

16.  Find  the  4th  term  in  the  series  2,  2^,  3 J, ... 

17.  Insert  four  harmonic  means  between  §  and  -j^^. 

18.  Find  the  two  numbers  between  which  12  and  9^  are 
respectively  the  geometrical  and  the  harmonic  mean. 

19.  If  a  body  falling  to  the  earth  descends  16^  feet  the 
first  second,  3  times  as  far  the  next,  5  times  as  far  the  third, 
and  so  on  ;  how  far  will  it  fall  during  the  /th  second  ?  How 
far  will  it  foil  in  /  seconds? 

20.  A  ball  falls  from  the  height  of  100  feet,  and  at  every 
fall  rebounds  one  fourth  the  distance  ;  find  the  distance 
passed  through  by  the  ball  before  it  comes  to  rest. 

21.  According  to  the  law  of  fall  given  in  Example  19, 
how  long  will  it  be  before  the  ball  in  Example  20  comes  to 
rest? 

Ans.  ^%  A/579  =  7-4805    seconds. 


122  ALGEBRA. 


SECOND    PART, 


CHAPTER   XII. 
FUNCTIONS   AND    THEORY    OF   LIMITS. 

218.  A  Variable  is  a  quantity  that  is,  or  is  sup- 
posed to  be,  changing  in  value.  Variables  are  usu- 
ally represented  by  the  final  letters  of  the  alphabet, 
as  X,  y,  2. 

The  time  since  any  past  event  is  a  variable.  The  length  of 
a  line  while  it  is  being  traced  by  a  moving  point,  is  a  variable. 
li X  represents  any  variable,  x^,  3  x^,  and  2  x^  —  4:1;  will  denote 
variables  also. 

A  Constant  is  a  quantity  whose  value  is,  or  is  sup- 
posed to  be,  fixed.  Constants  are  usually  represented 
by  figures  or  by  the  first  letters  of  the  alphabet. 
Particular  values  of  variables  are  constants,  and  they 
are  often  denoted  by  the  last  letters  with  accents,  as 
x'y  y,  x",  y'K 

The  time  between  any  two  given  dates  is  a  constant,  as  is 
also  the  distance  between  two  fixed  points.  Figures  denote 
absolute,  and  letters  denote  arbitrary  constants. 

219.  An  Independent  Variable  is  one  whose  value 
does  not  depend  upon  any  other  variable. 


FUNCTIONS.  123 

A  Dependent  Variable  is  one  whose  value  depends 
upon  one  or  more  other  variables.  A  dependent 
variable  is  called  a  Function  of  the  variable  or  varia- 
bles upon  which  it  depends. 

If  the  radius  of  the  base  of  a  cylinder  is  an  independent  va- 
riable, and  the  aUitude  is  always  four  times  the  radius,  then  the 
altitude  is  a  function  of  the  radius,  and  the  surface  and  volume 
are  different  functions  of  both  the  radius  and  the  altitude. 

The  expressions  ax^,  x*  —  ex,  a"^,  represent  functions  of  ;ir. 

If  in  any  equation  between  x  and  j/,  we  regard  x  as  an  inde- 
pendent variable,  then  j/  is  a  function  of  x.     Thus,  if 
y  =  2x''^  -{-  X  —  6,  and  x  increases  ;  then  when 
^=-4,  -3,   -2,   -I,       0,        I,    2,      3,     4,  ... 
jy=    22,       9,       0,   -s,   -6,   -3,   4,    15,   30,  ... 

Here  while  x  increases  from  —  4  to  —  i,  j  decreases  from  22 
to  —  5 ;  while  x  increases  from  —  i  to  i ,  y  first  decreases  and 
then  increases  ;  and  while  x  increases  from  i  to  4,  j/  increases 
from  —  3  to  30. 

220.  Functional  Notation.  The  symbol  /(x),  read 
"  function  of  x"  is  used  to  denote  any  function  of  x. 
When  several  dififerent  functions  of  x  occur  in  the 
same  discussion,  we  employ  other  symbols,  as  /'  (,i'), 
F(x)^  <!>  (;r),  which  are  read  "/"  prime  function 
of  x"  "  large  F  function  of  x,"  ''  </>  function  of  :r," 
respectively. 

The  symbols  f(a),  /(2),  /(r),  f(c  +  ^),  repre- 
sent the  values  of /(;r)  for  x  =  a,  2,  2,  c  +  d,  respec- 
tively. 

Thus,  if  /(x)  =x^  +  x,  /(a)  -a^-j-a,  f{2)  =  10, 
and  /(^  +  ^)  =  ^  +  3 ^^^+  3  ^^^'^  +  ci^  +  c  +  d 


124  ALGEBRA. 

Since /(;ir)  denotes  any  function  of  x,y=f{x) 
represents  any  equation  involving  x  and  j/,  when 
solved  for  y,  .  • 

221.  The  symbol  ^,  read  "factorial  ?2,"  denotes  the 
product  of  the  first  n  whole  numbers ;  that  is, 

1^=  I  •  2  .  3  .  4 n. 

Thus,    [3=1.2.3  =  6;  14  =  1-2.3.4  =  24. 

EXERCISE  19. 

In  the  first  three  examples  the  student  should  carefully 
note  \iQ'N  f  {x)  changes  as  x  increases. 

1.  /(.r)  =  5  a:^  -  3  ^  +  2  ;  find  /(-3),/(-  2),/(_  i), 
/(O),  /(i),  /(2),/(3)>  /(4),  /(5),  /(6). 

2.  f{x)  =  4.x^  —  x*+  2X—  IT,  find  /(—  2),  /(—  i), 
/(O),  /(I),  /(2),  /(3).  /(4),  /(5),  /(6)- 

3-  fix)  =x'  +  x'  +  2;  find  /(-  4),  /(-  3),  /(-  2), 
/(-  i),  /(-  0-3),  /(-  0.2),  /(O),  /(.),  /(2). 

4-  ^(x)  =x^+  sx;  find  /"(^j  +  2),  i^(a:i  +  «),  7^(5  x^), 

5-  -/^  (•^)  =^^  +  4^  +  3;  find  -^  (3  -^i).  -^  (*i  +  '^). 
^(^1-5),   fie-,  a). 

6.  fix)  =  («  +  ^)'"  ;  find  /(O),  /(i),  /(2),  fib),  fiz). 
7-  /W  =  <J',-  find/(0),/(i),/(2),/(3),/W,/(»«-«). 


THEORY  OF   LIMITS.  12$ 

Verify  the  following  identities  : 
8.    [5  =  120;  17=5040;  19  =  362880. 

18  In  ^       |9  |« 

10.    9  .  8  .  7  .  |6  =  [9  ; 

n{n  —  1)  («  —  2)  . . .  («  —  r  4-  i)  \n  —  ^  =  [«. 


II. 


9-8.  7-6_    |9 

|3         "^[5^ 
«(«—  i)(«  — 2)  ...(//—  r+  i)  _ 


l£  ~lltl 


r 


THEORY    OF    LIMITS. 

222.  Limit  of  a  Variable.  When,  according  to  its 
law  of  change,  a  variable  approaches  indefinitely  near 
a  constant,  but  can  never  reach  it,  the  constant  is 
called  the  Limit  of  the  variable.  A  variable  may  be 
always  less,  always  greater,  or  alternately  less  and 
greater  than  its  limit. 

If  a  regular  polygon  be  inscribed  in  a  circle,  and  another  be 
circumscribed  about  it,  and  if  the  number  of  their  sides  be 
doubled  again  and  again,  the  area  of  each  of  these  polygons 
will  approach  indefinitely  near  to,  but  can  never  equal,  the  area 
of  the  circle,  which  is  therefore  their  common  limit.  The  area 
of  the  inscribed  polygon  is  always  less  than  its  limit,  while 
that  of  the  circumscribed  is  always  greater. 

The  limit  of  the  perimeters  of  each  of  these  polygons  is  evi- 
dently the  circumference  of  the  circle.     The  variable  difference 


1 26  ALGEBRA. 

between  the  area  of  the  circle  and  that  of  either  polygon  con- 
tinually decreases,  and  evidently  approaches  zero  as  its  limit. 

/ 1  \  "  I 

If  ;/  increases  without  limit  (-)  ,  or  —  ,    approaches   zero 

as  its  limit ;  for  by  increasing  /z,  —  can  be  made  as  small  as  we 
please,  but  it  cannot  be  made  zero. 

223.  Corollary  i.  The  difference  between  a  va- 
riable and  its  limit  is  a  variable  whose  limit  is  zero ; 
that  is,  if  a  is  the  limit  of  x,  the  limit  of  a  —  x  is  zero. 

When  near  its  limit,  a  variable  whose  limit  is  zero 
is  called  an  Infinitesimal. 

224.  Corollary  2.  A  variable  cannot  approach 
two  unequal  limits  at  the  same  time. 

For  if  so,  in  approaching  one  of  these  limits  the 
variable  would  evidently  reach  a  value  intermediate 
between  the  two  unequal  limits,  and  then  it  would 
recede  from  one  of  them  while  it  approached  the 
other. 

225.  Corollary  3.  If  the  limit  of  v  is  zero,  the 
limit  of  cv  is  zero  also,  and  therefore  the  limit  of 
ca  —  cv   is  cay  c  and  a  being   any  constants. 

For  however  small  k  may  be,  we  may  make  v<ik-T-Cy 
whence  cv  <k ;  that  is,  cv  can  be  made  as  small  as 
we  please,  but  It  cannot  be  made  zero,  since  v  cannot; 
hence  the  limit  of  ^^'  is  zero. 

Again,  ca  —  cv  evidently  approaches  as  near  to  ca 
SiS  cv  does  to  zero  ;  hence  the  limit  of  c  a  —  c v  is  c a. 


THEORY   OF   LIMITS.  12/ 

Notation.  The  sign,  ^,  denotes  "approaches  as 
a  limit;"  thus,  x  =  a  is  read  '' x  approaches  a  as 
its  Hmit." 

The  Hmit  oi  x  is  often  written  briefly  It  {x). 

226.  If  two  variables  are  always  equal  and  one  ap- 
proaches a  limit,  the  other  approaches  the  same  limit; 
that  is,  if  y  =  ^.y  and  x  —  a,  then  y  —  a. 

For  evidently  if  x  approaches  indefinitely  near  to 
a,  but  cannot  reach  it,  then,  since  y  =  x^  y  also  must 
approach  indefinitely  near  to  a,  but  cannot  reach  it. 

227.  If  two  variables  are  always  equal  and  each 
approaches  a  limit,  their  li7nits  are  equal ;  that  is,  if 
y  =  X,  a7td  X  ==  a,  attd  y  —  b,  theft  b  =  a. 

l{  y  =  X,  and  x  =  a,  then,  by  §  226,  y  =  a.  But 
y^  by  whence  b  =  a,  since,  by  §  224,  y  cannot  ap- 
proach two  unequal  limits  at  the  same  time. 

228.  The  limit  of  the  product  of  a  constant  a7id  a 
variable  is  the  product  of  the  constant  and  the  limit 
of  the  variable ;  that  is,  if  It  (x)  =  a,  //  (c  x)  =  c  a. 


Let 

V  ~  a  —  x; 

then 

It  {v)  ^\t(a-x)  =  0, 

§§  227,  223. 

and 

cx^=ca  —  cv. 

Hence         \\.{cx)  =  \i(ca  —  cv)  =  ca.      §§227,225. 


128  ALGEBRA. 

229.  The  limit  of  the  variable  sum  of  a  finite  num- 
ber of  variables  is  the  sum  of  their  limits ;  that  is,  if 
It  {x)  =  a,  //  (y)  =  b,  It  (z)  =  c,  etc.  to  n  variables  ; 
the?i  ,   ,  ^  , 

For  let     i\  =  a  —  x,  V2  =  b  —y,  v^  —  c  —  z,  . . . ; 
then       It  (z/J  :=  0,   It  {y^  =  0,  It  (z^  =  0,  ..., 

and  x^-y-\-z-\ —  (a  +  b  +  c-] )  —  {'Vi  +  V2  +  Vs+.")r 

Hence 

\t{x  +  y  +  z+...)=:]tl(a  +  b  +  c-+..-)-(v,  +  7>2  +  Vs+-")^. 

Now  however  small  k  may  be,  each  one  of  the  n 
variables,  Vi,  V2,  Vs,  ••• ,  can  be  made  less  than  k  -^  n ; 
therefore  their  sum  can  be  made  less  than  k ;  hence 

lt(z;i  +  2^2  + ^'3  +  •••)  =  0. 

Hence   It  (^+7 +  2+ ...)  ==  ^  + /^  +  ^  4- ••• 

230.  The  limit  of  the  variable  product  of  two  or 
more  variables  is  the  product  of  their  limits;  that  is, 
if  lt(x)  =  a,  and  It  (y)  =  b,  then 

\t(xy)  =ab  =  \t(x)  \t  (y). 

Let  Vi  =  a  —  X  and  7'2  =  b  —y; 

then  x  =  a  —  Vj,  y  =  b  —  v^y 

and  X y  =  a  b  —  a  V2  —  b  7^1  +  z^i  z'2  • 

.*.  It  (xy)  =\\.{ab  —  av^—  It  {b  v^  +  It  {v^  v^    §  229. 
=  ab^\i{x)  It  {y). 

In  like  manner  the  theorem  is  proved  for  n  variables. 


THEORY   OF  LIMITS.  1 29 

231.  The  limit  of  the  variable  quotient  of  two  varia- 
bles is  the  quotient  of  their  limits ;  that  isy 

\\.{x-ry)  =  \\.{x)-^\\.{y). 

Let  z=^x-^y,  Qxx^^yz; 

then  \\.{z)  =  \i{x-^y), 

and  lt(^)=:lt(>'0)=lt(j)lt(2).           (i) 

Hence  It  (^)  =  It  (^) -- It  (/).                     (2) 

Whence  It  {x  ^  y)  ^  It  {x)  H-  It  {y). 

Remark.  The  demonstration  given  above  fails, 
and  the  theorem  is  not  true,  when  It(^),  or  the  Hmit 
of  the  divisor,  is  zero;  for  then  we  cannot  divide  (i) 
by  lt(j)  to  obtain  (2). 

232.  Whc7i  the  product,  quotient,  or  sum  of  two  or 
more  variables  is  equal  to  a  constant,  the  product,  quo- 
tient, or  sum  of  their  limits,  is  equal  to  the  same 
constant, 

(i.)   Let    xy  =  m  ;  then  xy  z  =  mz. 

.-.  \\.(x)\\.{)^\t{z)  =  m\l(z). 
.'.  It  (x)  It  (>')  =  m. 
(ii.)   Let   X -^  y  =  m ;  then  x=7ny. 

.'.  It  (.v)  =  m  It  (y),  or  It  (x)  -4-  It  (y)  =  m. 
(iii.)   Let  x+y+z  +  '"  =  m; 

then  y  +  z  -{-'•'  =  m  —  X. 

.-.  It  (y)  +  It (z)  +  ...  =  m  —  \t  (x). 
.-.  \t(x)  +  \t(y)  +  \t(z)  +  ...  =  m. 


I30  ALGEBRA. 

233.    If  a  is  finite,  and  ;r  =  0,  then  the  fraction  - 

X 

will  numerically  increase  without  limit. 
\{x  increases  without  Hmit,  then  -  =  0. 

X 

A  variable  that  increases  without  limit  is  called  an 
Infinite.  The  value  of  an  infinite  is  denoted  by  the 
symbol  oo  ,  and  ;t:  —  x  is  read  '' x  increases  without 
limit,"  or  '' x  is  infinite."  With  this  notation  the  two 
statements  made  above  may  be  written  as  follows : 


If  jc  =  0,  then  -  r=  DO  ; 

X 

if  ^  =  00,  then  -  =  0. 


Example   i.     Find  It  ^^_. — -ttq.^  if  ;»r=». 


*ExAMPLE  2.     If  ;r=  0,  and  «  >  0,  then  It  (^^)  =  i. 
Let  a^  \,  X  positive,  and  k  a  positive  number  as  small  as 
we  please  ;  then,  as  i  -^  ;f  =  », 

,'.     1  +  k  >  a''  >  I,  or  It  (^^)  =  I. 

The  proof  is  similar  for  a  <  i.  For  x  negative  we  have 
^z^  =  (i  -f  a)"",  in  which  —  x  is  positive. 

In  this  example  x  is  commensurable,  and  only  the  positive 
real  values  of  «*  are  considered. 


THEORY   OF  LIMITS.  131 


EXERCISE  20. 


1.  Prove  It  (at")  =  [lt(A:)]^ 

Lt  (x'')  =\t(x  -  X  '"  to  n  factors) 

=  It  (x)  .  It  (^)  ...  to  «  factors 

=  [it(^)]-. 

2.  Prove   It  U~'/  =  [It  (or)]". 

Let  x*"  =  z;  then  xf"  =  z",  etc. 

3.  Prove  lt(^y  =  /;/  -H  [It  (x)^. 

If  It  (x)  =  a,  It  ( j)  =  ^,  It  (2)  =  c,  and  It  (z;)  =  0,  find 

4.  Lt  (^^y-j  +  axz). 

5.  Lt (a;2 ^?  -{■  mx z^  +  nxzv). 

,i_ 

xy  ■\-  nx^  ■\-  my 


f  x"^ y^  ■\-  ms^  -\-  nzv\ 
\  xy  ■\-  nofi  ■\-  my  v  J 


Find  the  limit  of  each  of  the   following  expressions,  (i.) 
when  ;«:  =  0,    (ii.)  when  ;c  =  ao  : 


8. 


2^*  +  4 


mx^ -\- nx^ -\- px -\- q' 

10,    (2  ^  -  3)  (3  -  5  ^) 

70:'^— 6x  +  4  ''^*    2;c^— I    ■     2  x^ 


I. 

+  9 

(3  + 

2  ;c«)  (a: 

-5) 

(4^= 

-9)  (I 

+  ^) 

I    — 

^'     .    I 

—  X 

132  ALGEBRA. 

234.  Vanishing  Fractions.  If  in  the  fraction 
ir^ — ^ —  we  put  X  —  a,  it  will  assume  the  inde- 
terminate   form   - .     A   fraction  which  assumes  this 

form  for  any  particular  value  of  x\  as  a^  is  called  a 
Vanishing   Fraction  for  X  =^  a. 

To  find  the  value  of  such  a  fraction  for  ;r=  «,  we 
find  its  limit  when  x  =  a. 

The    limit  of  ^ o when  ;ir  =  ^  is  often 

x^  —  a^ 

limit    \x^  -\-  ax  —  2  a'^l 

written        .        o 2 • 

X  =  a  I       x^  ~  a^       \ 

^  „.    ,    limit   \x^  +  a  X  —  2a^'\ 

Example.    Find       .        — —^ ^ . 

X  =  a  \_       x^  —  a^       J 

Here  the  limit  of  the  divisor,  jr^  —  a^^  is  zero,  and  we  cannot 

apply  §  231.     But  as  long  as  x  is  not  absolutely  equal  to  a,  we 

may  divide  both  terms  of  the  fraction  by  x  —  aj  hence 

x"^  ^  a  X  —  ia^      X  -\-  1  a 


x^-a'^  x  +  a 

limit   [x"^-^  ax-2  ^^  _   limit   p  +  2  fzl  _  3 
x=  a\         x^  —  a^         \~  x  =  a\    X  -{-  a  \       2 


*235.  Incommensurable  Exponents.  If  di  ts  positive 
and  m  is  incomtnejisurab  ^e^  the  positive  real  value  of 
a""  is  the  limit  of  the  positive  real  value  of  a"  when 
X  ~  m. 

Let  X  and  jj/  be  commensurable,  and  let 

X  <  m  <  y,   \i{x)  =  m  =  It  {y). 
Then,  if  ^  >  i,  we  assume  as  axiomatic  that 

a""'  <  a^,   or   «""  —  a""  <,  a^  —  a". 


THEORY   OF  LIMITS. 


133 


But       a^  —  a''  =  (f{a^-''  —  i)  =  0.     Example  2,  §  233. 

/,     oT  —  It  (t?')    when   a  >  \    and   x  =  m. 

The  proof  is  similar  when  a  <  i. 

This  proof  applies  also  when  m  is  commensurable. 

Example.  Prove  the  laws  in  §  70,  when  m  and  n  are  in- 
commensurable, a  and  b  being  positive. 

Let  X  and  J  denote  any  two  commensurable  fractions  whose 
limits  are  m  and  n  respectively  ;  then 

It  («*  a>)  =  It  (^*+>)  =  «'"  +  ", 
also  It  {a^ay)  =  It  («0  •  It  (^-»)  =  a'^a\ 

Similarly  the  other  laws  are  proved. 

Remark.  Except  when  the  limit  of  a  divisor  is 
zero,  the  limit  of  each  function  considered  in  this 
chapter  is  the  result  obtained  by  substituting  for  the 
variables  their  respective  limits. 


EXERCISE  21. 


Find 

Limit  r^r"  —  il 

X  =  iix  —  I  y 

Limit     [x^  -}■  1*1 
^  =  —  I  Ix^  —  I  J  * 


Limit 
X  =  a 


(x  -  a)^ 


Limit  [x*  —  ^*1 

4.         .        —    . 

x^=alx  —  aj 

Limit   rx^  —  a^l 
X  =  al  X  —  a  1 

\/a  —  \/x 


Limit 


X  —  a  I  ^a  —  X  . 


In  Example  6  rationalize  the  numerator. 


134  ALGEBRA. 


CHAPTER    XIII. 
DERIVATIVES. 

236.  The  amount  of  any  change  (increase  or 
decrease)  in  the  value  of  a  variable  is  called  an 
Increment.  If  a  variable  is  increasing,  its  increment 
IS  positive ;  if  it  is  decreasing,  its  increment  is  neg- 
ative. An  increment  of  a  variable  is  denoted  by 
writing  the  letter  A  before  it ;  thus  A  x,  Ay,  A  z, 
denote  the  increments  o{  x,y,  z,  respectively.  Hence 
if  x^  denote  any  value  of  ;r,  x^  -\-  A  x  denotes  a 
subsequent  value  of  x.  \{  y  is  a  function  of  x, 
and  y  =  y^  when  x  —  x^ ;  then  y  =y  +  Ay^  when 
X  =  x^  +  A  X. 

237.  The  Derivative  of  a  function  is  the  limit  of 
the  ratio  of  the  increment  of  the  function  to  the 
increment  of  the  variable  as  the  increment  of  the 
variable   approaches  zero   as   its    limit. 

If  ^  is  a  function  of  x,  the  derivative  of  y  with 
respect  to  x  is  often  denoted  by  D^y.  Hence  by 
definition,  we  have 


limit 


t^-^-y  w 


DERIVATIVES.  135 

Example  i.    Given  j/  =  ax^  +  cx-{-l>,  to  find  Z>^^. 
Let  x^  and  y  be  any  two  corresponding  values  of  x  and  jy 

then  y  =  ax'^  +  cx'  +  d.  (i) 

When   X  =  x'  +  A X,   y  =/  +  Ajy  hence  we  have 

y  +  A_y  =  a(x'  +  Axy  +  c(x+  Ax)  +d 

=  ax'^  +  2ax'Ax  +  aAx^  +  ex*  +  CAX+  d.     (2) 

Subtracting  (i)  from  (2),  we  obtain 

Af=  2ax'Ax+aAx^  +  cAx, 

Dividing  by  A  x,  we  have 

— ^  =  2ax*  +  aAx  +  c, 
Ax 


==  2ax'  +  c. 


Hence,  as  x'  is  any  value  of  x^  we  have  in  general, 
£>xy -D^{ax'^  ^cx-\-b)  =  2ax-Vc. 

In  case  J/  is  a  linear  function  of  x,  as  when  j  =  ^;r, 
by  the  method  above,  or  by  inspection,  we  find  that 
Ay  -^  Ax  =.  a,2i  constant. 

Here  the  ratio  of  A^  to  A,r,  being  a  constant,  cannot 
approach  a  limit  as  A;ir  =  0.  In  this  case  the  deriva- 
tive o{ y  is  the  constant  ratio  of  Aj^  to  A;r;   that  is 

D^y  =  D,(ax)  =  a, 
l{ a=  I,  we  have  D^^y  =  D^x  —  i. 

Thus,  Z)^  (2  :r)  =  2  ;  D^{tix)  =  n\  D^{^cx)  —  -c\  D^x=i, 


136 


ALGEBRA. 


238.  The  de7'ivative  of  a  function  is  positive  ornega- 
tive  according  as  the  function  increases  or  decreases^ 
when  its  variable  increases ;    and  conversely. 

For  \i  y  increases  when  x  increases,  Ay  and  Ax 
have  hke  signs;  hence,  from  [i]  of  §  237,  i>^_;/  is 
positive.  \i  y  decreases  when  x  increases,  Ay  and 
Ax  have  unHke  signs;  hence  Dj,y  is  negative;  and 
conversely. 

EXERCISE   22 

Find  the  derivative  of  _>',  if 


\.  y  ■=  x"-. 

2.  y  =  a  x^  -\-  b, 

3.  y  —  c  x^  —  a  X, 
4'  y  =  x\ 
5.  y=zcx^  —  ax\ 

Note.    To  the  expression 


x""  —  cx^  -\-  4.x. 

4 


6.  y 

7.  y  =  ax 

8.  y  =  cx'  —  bx\ 

9.  y  =^  a  -^  X. 

10.   y  =  X  -^  (a  —  x) 


limit 


0  Lax] 


Ax  =  {)  LA:rJ^^^  cannot  apply  the 
principle  of  §  231,  for  the  limit  of  the  divisor,  Ax,  is  zero. 


239.  Geometric  Illustration  of  a  Derivative, 
ceive  a  variable  right  triangle 
with  constant  angles  as  gene- 
rated by  the  perpendicular  BC 
moving  uniformly  to  the  right. 
Let  X  denote  the  variable  base, 
2  ax  its  altitude,  and  y  its  area; 
then,  by  geometry,  y  =  ax^. 

Let  A  B  he  any  value  of  ;ir,  and 
let  Ax  =  B  H; 

then  t^y  =  zxediBHNC.  a 


Con- 


DERIVATIVES.  137 

Let  MS  join  the  middle  points  oi  B H  and  C N; 
then,  by  geometry, 

'QiQ3iBHNC=BHx  MS. 

Ay      zx^z.BHNC      BHxMS       . ,  „  ,, 

'-'-£=         BH        -        BH        =^^'         W 

As  A  ,r  z^  0,  MS  =B  C ;  hence,  from  (i)  we  have 

limit     rM'l  limit      \ms\  =  BC; 

,\  D^y  —  BC=2ax,  §237. 

or  D^  {a  x^)  =  za  x. 

240.  The  sign  of  the  operation  of  finding  the  de- 
rivative of  a  function  with  respect  to  x  is  D^.  Thus, 
D^  in  D^{x^)  indicates  the  operation  of  finding  the 
derivative  of  ,r^,  while  the  whole  expression  Z)^  (x^) 
denotes  the  derivative  of  x^. 

We  could  obtain  the  derivative  of  any  function  by 
the  method  of  §  237 ;  but  in  practice  it  is  more  expe- 
dient to  use  the  following  general  principles: 

241.  Tke  derivative  of  equal  functions  of  the  same 
variable  are  equal. 

\{u  2CCiAy  are  equal  functions  of  ;r,  we  are  to  prove 
that  D,  u  =  D,y, 

A«      A^ 

If  u=y,^u  =  ^y',    '•^^  =  ^ 


138  ALGEBRA. 

Tj.  limit     r^'^l  liii^it      [-^yl  n 

Hence,     ^  U^^  1^  .  §227. 

.-.  Z>^u  =■  D^y.  §  237. 

242.  The  derivative  of  the  product  of  a  constant  and 
a  variable  is  the  product  of  the  constant  and  the  deriva- 
tive of  the  variable. 

We  are  to  prove  that  D^  {ay)  —  a  D^y^  in  which  y  is 
some  function  oi  x. 

Let  u—  ay,  and  let  x  denote  any  value  of  x,  and 
y  and  t/  the  corresponding  values  of  j^  and  u  respec- 
tively; then 

a  =  ay',  (i) 

When x  —  x'-^Ax,  then yz=iy'  -\-  Ay^  and  u  —  u'-\-Au\ 
hence     u'  -\-  Au  —  a  (/  +  Ay)  =  a/  +  a  Ay,  (2) 

Subtracting  (i)  from  (2),  we  have 

Au  ■=  a  Ay. 

,    Au  _     Ay 

Ax~     Ax 

Ajv=OLA^J       Ajt:  =  OLA::cJ 

limit     r^  J^      R  ^^Q 
=  «  •  __^    .     §  228. 

Aa:=  OLa^vJ 
.-.  D,u  =  aD,y.  §  237. 

Hence,  as  x^  is  any  value  of  ;t-,  we  have  in  general 
D^  u  =  D^  (ay)  —  a  D^y, 
Thus,  A  (3  ^  ^;r8)  =  3  «  3  .  Z7^  {x^y 


DERIVATIVES.  1 39 

243.    The  derivative  of  a  constant  is  zero. 

If  ^  is  a  constant,  A(a;  =  0  ;  .'.  —  =  0,  .*.  D^a  =  ^. 

Ax 


244.  The  derivative  of  a  polynomial  is  the  algebraic 
sum  of  the  derivatives  of  its  several  terms. 

We  are  to  prove  that  D^{v  ^ y  -\-  z  -\-  a^  =  D^v 
-f  D^y  +  D^z,  in  which  v,  y,  and  z  are  functions  of  ;r. 

Let  n  —  v  +y  +  z  +  a,  and  let  y  represent  any  value 
of  ;r,  and  v\y,  z,  and  t/  the  corresponding  values  of 
V,  y,  z,  and  «,  respectively ;  then 

u'  =  v'  +y  +  z'  +  a.  (i) 

When  X  =  x^  +  Ax,  then  v  =  v'  +  Av,  y=y  +  Ay, 
z  =  si  +  Az,  and  u=  u^  -\-  Au;  hence 

u'  +  Au  =  v'  +  ^v  -{-/  +  ^y  +  z'  +  Az  +  a.     (2) 

Subtracting  (i)  from  (2),  we  have 

Au  =  Av  +  Ay  -\-  ^z. 

.    Au  _Av       Ay       A  z 
'  '  Ax~  Ax       Ax       Ax 

.       limit     [Aw]^     limit     [^  ,   M'  ,   ^1      §227 
'*A:^  =  0LajcJ       A^  =  0LA^       A^       Ax\ 

__     limit    r^^l  ,       limit    TAjH  limit     ["A  si 


I40  ALGEBRA. 

Hence,  as  x!  is  any  value  of  x,  we  have  in  general 

=  3a'D  (x^)  -2n(x^)  +  sa. 

245.  T/ie  derivative  of  the  product  of  two  variables 
is  the  first  into  the  derivative  of  the  second^  plus  the 
second  into  the  derivative  of  the  first. 

We  are  to  prove  that  D^  {y  z)  —y  D^2  -\-  zD^y,  in 
which  _y  and  s  are  functions  of  ;ir. 

Let  u—yz,  and  let  x  represent  any  value  of  ;r,  and 

y,  2^,  and  u'  the  corresponding  values  of  y,  z^  and  Uy 

respectively;  then 

u'^y'^.  (i) 

When  x  =  x'  ^  ^x^  then  ^  =  /  +  A  j,  2  =  s'  +  A  2:, 

and  u^u'  -^  ti.u\  hence 

//+  A ?/=(/+  ^y) (/+  A ^)=//+y  A2  +  /Aj;4-  A0 Aj.  (2) 

Subtracting  (i)  from  (2),  we  obtain 

A«=yA0  +  '2;'A_y  +  A^r  A  J. 
.    A  u 


AJt:       -^   A^       ^  ^  A^ 


limit 
A  j; 


lit     TA^/l  limit     r    ,A^       (^,       ^^^Aj-l 

=  ^^"^^^  [y-l+  ^^"^^'  r(.  +  A.)^i^l 

A.t=OL      AjcJ      A.t=OL^  A^J 

=y  ^i"^it  r^^l+  ^i^^t  [z  +  Asl .  ^'"^'^  F^l 

•^   Ax=OLA:rJ     A^  =  OL  J      A:^  =  OLa;<;J 


/.  D^u  =y'D,z  +  2'  Dxy^ 


DERIVATIVES.  I4I 

Hence,  as  x'  is  any  value  oi  Xy  we  have  in  general 
D,u  =  D,  {yz)  =yD,z  ^zD.y. 

246.  The  derivative  of  the  product  of  any  number  of 
variables  is  the  sum  of  the  products  of  the  derivative  of 
each  into  all  the  rest. 

We  are  to  prove  that  D^  {vy  z)—yzD^v-Vvz  D^y 
+  vyD^z, 

Let  u  =  vy; 

then  Dx  {vyz)  =  A  («^)  §  241. 

=  zD,u  +  uZ>xZ  §245. 

=  zDj,  {vy)  +  vy  £>xZ 

=  yzD^v -\- vzJDj^y -\- vyZ>^z.       §245. 

In  a  similar  manner  the  theorem  may  be  demon- 
strated for  any  number  of  variables. 

247.  The  derivative  of  a  fraction  is  the  denominator 
into  the  derivative  of  the  numerator ^  minus  the  numc- 
7'ator  into  the  derivative  of  the  denominator^  divided 
by  the  square  of  the  denominator. 

We  are  to  prove  that  A  \-]= J! — ^'  ^^ 

which  y  and  z  are  functions  of  x. 

Let  «  =  -^ ;  then  uz=y. 

z 

.*•  u D^z  ■\-  zD^u  =  D^y.  §245. 


142  ALGEBRA. 

D^y  —  uD^z  _ 


n^y-^D^z 


.'.n^u=. 


248.    By  §247,  A  (^l) 


_zD,y-yD,z 

""  z^ 

z  D^a  —  aD^z 


^.  §  243. 


Hence,  the  derivative  of  a  fraction  with  a  constant 
numerator  is  minus  the  numerator  into  the  deriva- 
tive of  the  denominator  divided  by  the  square  of  the 
denominator. 


249. 


By§.43.A@  =  A(i.)=>.,  =  ^. 


250.  The  derivative  of  a  variable  base  affected  with 
any  constant  exponent  is  the  product  of  the  exp07ient 
into  the  base  with  its  exponent  diminished  by  one,  into 
the  derivative  of  the  base. 

(i)  If  the  exponent  is  a  positive  integer,  ^s  m\ 
then  D^  {f)  —  D^iz-z-z  -"  \.o  m  factors) 

=  2"*-'  D^z  +  z""'^ £>^z  +  '•'  to  m  terms 

=  mz'''~^  Z>^z. 


DERIVATIVES.  1 43 


1ft 
(2)  If  the  exponent  Is  any  positive  fraction,  as  — ; 


let  u  —  z"\  then  «"  —  z"\ 

.'.  n  ti""-^  D,  u  =  m  z'"-'  D,z, 

.*. D^u  =  -  — —  D^z= D^ z 

m 

(3)  If  the  exponent  is  any  negative  quantity,  as  — ;/ ; 
then^-"=  -.  .  (0 

By  §  248,  we  obtain  from  (i) 

—  —  nz-"-^D,z. 
Thus,Z?.(;r8)  =  3;i-2;  Z?,  (:rt)  =  f  x?  ;  Z?  (x-0  =  -2^-3. 

251.  By§250,Z>.(V'<^)  =  A-(^)  =  J^~^A^=J^- 

Hence,  /^^  derivative  of  the  square  root  of  a  variable 
is  the  derivative  of  the  variable  divided  by  twice  the 
square  root  of  the  variable. 

252.  The  general  symbol  for  the  derivative  oifix) 
is/'  (.r) ;  that  is  A  [/W]  =/'  {x). 


144  ALGEBRA. 

EXERCISE  23. 

Find  the  derivative  of, 

I.     X^  +  SX  +  2X^ 

D^{x^  +  8^  +  2x')  =  I^.(x')+DX8x)+jD^(2x')  §  244. 
=  3x^  +  8  +  4x.  §250. 

2.  /(x)  =  3  ax^  —  ^nx  —  8  m. 
fix)  ~D,{3ax''-^7ix-8ni) 

=  A(3  «^')  -  ^.(5  «^)  -  A(8  m) 
~  6  ax  —  5  «. 

f(x)=  loax-gPx''  —  4adx\ 

5.  y  =  ax2+^x^-\rc. 

6.  /(x)  =  (d  +  ax^)i. 

7'  y=(i  +  2X'')  (i  +  ^  x^). 

In  Example  8 

D,y  =  (^  +  ^)^-(^+^^)-(^  +  a^)DAr+^)  ^    b  -  a^ 

10.  y^{a^x)^Jl^^.  II.  /(^)  =  _1^. 


DERIVATIVES.  I4S 


13-  /(*)  =  v/f 


14-  /W  = 5  •  '9-  /W  = 


(i-*' 


(1  +  ^) 


15.  ^  =  — ^=z=7^  •  20.  f{x)  =  ^ax  +  ^/c^x^, 

16.  /(^)  =  ^4x~J*       ^^-  y  =  2/^. 

^  X  -\-  X 

In  Example  21,  Z);t(j^)  =  D^iip x),t\.z. 


22.   rt^_y^  -f  ^2  ^2  _  ^2  ^2^  23.    2  .v_>'^  —  ay^  =  x^. 

24.  Prove  the  theorems  of  §§  246,  247  by  the  general 
method  used  in  the  previous  articles. 

25.  Assuming  the  binomial  theorem,  prove  the  theorem 
of  §  250  by  the  general  method. 

253.  Successive  Derivatives.  Since  f'{x)y  the  de- 
rivative of /(a'),  is  in  general  another  function  of  jr, 
it  can  be  differentiated  the  same  asy(.i').  The  deriva- 
tive oi  fix)  is  called  the  Second  Derivative  of  the 
original  function /(jr),  and  is  denoted  hyf'ix).  The 
derivative  c>{  f'\x)  is  called  the  Third  Derivative  oi 
f\x),  and  is  represented  hy  f"'{x) ;   and  so  on.    f"{x) 


146  ALGEBRA. 

represents  the  nth.  derivative  of/(;r),  or  the  derivative 

Thus,  if       /(x)  =  x^,         then      /'  (x)  =  4  x», 
f"  (;f )  =  1 2  x\  f"  {x)  =  24  X, 

/'M^)  =  24,  /^W^O. 

/'W>  /"W*  /'"(-^).  •••,  /"W  are  called  the  5?/^- 
cessive  Derivatives  of  /(:i'). 

EXERCISE  24. 

Find  the  successive  derivatives  oi  f{pc)^  when 

1.  /(^)  =  ^^  +  2  :r2  +  ;r  +  7-       3-  /W  =  (^  +  ^)^ 

2.  fix)  =,cx^  ^  ax"  ^b.  4.  /(x)  =  (^  +  xy. 
5.  /(^)  =  ^o+  A-^  +  ^2^'+  -43-^'+  -^4-^'+  ^5^'+  ^,^T«. 

254.  Continuity.  A  variable  is  Continuous,  or  varies 
contimwiLsly,  when  in  passing  from  one  value  to  an- 
other it  passes  successively  through  all  intermediate 
values.     Otherwise  it  is  disconti^mous. 

A  Continuous  function  is  one  that  varies  continu- 
ously, when  its  variable  is  continuous.  Hence  ^  is  a 
continuous  function  of  x,  if  for  each  real  finite  value 
of  ;ir,  y  is  real,  finite,  and  determinate,  and  \{  A y  =  0 
when  A  X  =  0. 

The  time  since  any  past  event  is  a  continuous  variable,  as  is 
also  the  length  of  a  line  while  being  traced  by  a  moving  point. 

The  velocity  acquired  by  a  falling  body,  and  the  distance 
fallen,  are  continuous  functions  of  the  time. 


DERIVATIVES.  1 47 

The  area  and  the  altitude  of  the  triangle  in  §  239  are  contin- 
uous functions  of  the  base. 

The  number  of  sides  of  a  regular  polygon  inscribed  in  a 
circle,  when  indefinitely  increased,  is  a  discontinuous  variable, 
as  is  also  the  perimeter  or  the  area  of  the  polygon.  For  each 
of  these  variables  passes  from  one  value  to  another  without 
passing  through  all  intermediate  values. 

In  general  i  -^  ;r  is  a  continuous  function  of  .ry  but  when  x 
increases  and  passes  through  zero,  i  ^  x  leaps  from  —  »  to  +  ao; 
hence  i  -f-  x  is  discontinuous  for  x—^. 

255.  Any  rational  integral  function  of  x  is  con- 
tinuous. 

Let  «  be  a  positive  integer,  and 

y  =  A^x"  +  A^x'—"^  H h  A„_^x  +  A,„ 

then  for  each  real  finite  value  of  x^  y  has  one  real 
finite  value,  and  only  one. 

Again,  if  by  the  method  in  Example  i  of  §  237,  we 
obtain  Ay  in  terms  of  Ax,  Ay  will  be  found  equal  to 
A  X  multiplied  by  a  finite  quantity ;  hence  J  j/  =  0 
when  Ax  =  0. 

Therefore  y,  or  A^x"  +  A^x"''^  +  *,.  +  A„,  is  a 
continuous  function  of  x. 

Thus,  if  J/  =  x^  +  2x'^  +  X,  (i) 

Aj  =  (3^^+  3^'^^  +  t:^^''^  +  4^  +  2  Ax  +  i)  A;t-.     (2) 

From  (i),/has  one  finite  real  value  for  each  finite  real  value 
of  .1',  and  from  (2),  Ay  =  0  when  A  :r  =  0.  Hence  x^  +  2x^  +  x 
is  a  continuous  function  of  x. 


148  ALGEBRA. 


CHAPTER  XIV. 

DEVELOPMENT   OF    FUNCTIONS   IN   SERIES. 

256.  A  Series  is  an  expression  in  which  the  suc- 
cessive terms  are  formed  by  some  regular  law.  A 
Finite  series  is  one  of  which  the  number  of  terms  is 
limited.  An  Infinite  series  is  one  of  which  the  num- 
ber of  terms  is  unlimited. 

257.  An  infinite  series  is  said  to  be  Convergent 
when  the  sum  of  the  first  n  terms  approaches  a  limit 
as  n  is  increased  indefinitely ;  and  the  limit  is  called 
the  Sum  of  the  infinite  series.  If  the  sum  of  the  first 
n  terms  of  an  infinite  series  does  not  approach  a  defi- 
nite limit  when  ;^  =  »,  the  series  is  Divergent^  and 
has  no  sum. 

Thus,  the  infinite  geometrical  series 

I  +  -  +  -  +o  +  ---  +  -^  +  --- 
248  2"-l 

is  convergent;  for  by  §211  the  sum  of  its  first  n  terms  ap- 
proaches 1-^(1—  ^),  or  2,  as  its  limit,  when  ;/  =  x. 

The  series  i  +  i  +  i  +  i+---is  divergent,  since  the  sum  of 
its  first  ti  terms  increases  indefinitely  with  ;/. 

The  series  i  —  i  +  i  —  i+---is  divergent;  for  the  sum  of 
its  first  n  terms  does  not  approach  a  limit,  but  alternates  be- 
tween I  and  0  according  as  n  is  odd  or  even. 


DEVELOPMENT   OF  FUNCTIONS   IN  SERIES.      1 49 

258.  To  Develop  a  function  is  to  find  a  series  the 
sum  of  which  is  equal  to  the  function.  Hence  the 
development  of  a  function  is  either  difi7iite  or  a  co7i- 
vergent  infinite  series. 

Thus,  the  development  of  the  function  {a  +  xy  is  the  finite 
series  a*  +  ^a^x  +  6a^x^  +  ^ax^  +  x*. 

259.  Development  of  Functions  by  Division. 

I  —  X** 

Example  i.   Develop  by  division. 

Dividing  i  —  jr"  by  i  —  ;r,  we  obtain 

I  -X** 


I  —X 


=  l  +  x  +  x^  +  x^+...x'*-K  (i) 


If  n  is  finite,  the  series  in  identity  (i)  is  finite  and  is  the  de- 
velopment  of  the  function  for  all  values  of  x. 


Example  2.    Develop  by  division. 

I  —  X 

Dividing  i  by  i  —  x,  we  obtain 

I 


I  —X 


=  i+x  +  x^  +  x^  +  ...  +  x»-'^  +  ...,        (l) 


in  which  x"-"^  is  the  «th  or  general  term  of  the  series. 

The  series  in  (i)  is  infinite,  and  hence  it  must  be  convergent 
to  be  the  development  of  the  function. 

If  X  is  numerically  less  than  i,  the  series  is  evidently  a  de- 
creasing geometrical  series,  of  which  the  first  term  is  i,  and  the 

ratio  xy  hence   by  §  211  the  sum  of  ;/  terms  approaches 

I  —  X 
as  a  limit  when  «  =  ». 


150  ALGEBRA. 

If  X  is  numerically  greater  than  i,  the  sum  increases  indefi- 
nitely with  nj  hence  the  series  is  divergent,  and  is  not  the  de- 
velopment of  the  function.     Thus,  for  or  =  2,  the  series  becomes 

I +  2  +  4  +  8+ 16  +  ..., 

while  the  function  equals  —  i. 

\i x=  I,  the  sum  of  the  first  n  terms  is  n,  and  therefore  the 
series  is  divergent. 

If  :ir  =  —  I,  the  series  becomes  the  divergent  series 

I  -  I  +  I  -  I  +  I  -I  +  ... 

Hence  the  series  in  (i)  is  the  development  of  only  for 

values  of  ;r  between  —  i  and  +  i. 

Example  3.    Develop  — —  by  division. 

I  +  X 

Dividing  ;r  by  i  +  A",  we  obtain 

-J^  =  x-x^  +  x^-x^  +  x^ +  (-i)«-i;f''  +  ... 

I  +  X 

Here  the  series  is  evidently  divergent  for  all  values  of  x 
except  those  between  —  i  and  +  i.  For  values  of  x  between 
—  I  and  +  I  the  series  is  a  decreasing  geometrical  progression 
of  which  the  first  term  is  x  and  the  ratio  is  —  xj  hence  the  sum 

of  n  terms  approaches  ■ as  its  limit,  when  ?i  =  00. 

Principles  of  Undetermined  Coefficients. 

260.  Undetermined  Coefficients  are  assumed  coeffi- 
cients whose  values,  not  known  at  the  outset,  are  to 
be  determined  in  the  course  of  the  demonstration  of 
a  theorem  or  the  solution  of  a  problem. 


DEVELOPMENT   OF  FUNCTIONS   IN   SERIES.      15I 

261.  If  ^0  =  Bo,  Ai  =  Bj,  A2  =  B2,  Ag  =  B3,  ...,  theft 
Ao  +  A,x  +  A2x2+  .  .  .  =Bo  +  B,x  +  B2x2+  .  .  .;(i) 
that  is,  if  in  tJie  two  members  of  an  equality  the  coeffi- 
cients of  the  like  powers  of  y.  are  identical^  the  equality 
is  an  identity. 

For  by  hypothesis  we  have  the  identities 

Aq  =  Bq,  Ai X  =  Bi X,  Azx'^  =  B2x\  •  • . 

Adding  these  identities,  we  obtain  the  identity  (i). 

262.  Conversely,  if 

A,  +  A,x  +  A^x^  +  ...  =  ^0  +  ^1^  +  ^2^'  +•••,  (i) 
then  Ao  =  Bo,  Ai  =  B^  .  »  -; 

that  isy  in  an  identity,  the  coefficiejits  of  the  like  powers 
of  the  variable  in  the  two  members  are  identical. 

For  Aq  and  ^0  ^^e  respectively  the  limits  of  the 
equal  varying  members  of  (i),  as  ;r  =  0 ;  hence 

A,  =  B,.  (2) 

Subtracting  (2)  from  (i),  and  then  dividing  by  ;ir, 
we  obtain 

A^  +  A^x  +  A^x"  +  ...  =  ^1  +  B^x  +  B^x^  +  ...   (3) 

l(  X  =  0,  from  (3)  we  obtain 
A,=B,. 

In  like  manner  we  may  prove 

A2  =  B2,  A3  =  Bs, . . . 


152 


ALGEBRA. 


263.    Development    of     Functions     by    Undetermined 
Coefficients. 


Example  i.    Develop 


I  -x^ 


Clearing  (i)  of  fractions,  and  for  convenience  writing  the 
coefficients  of  the  like  powers  of  x  in  vertical  columns,  we 
obtain 


x^^A^^A^ 


x  +  A^ 

x'+A^ 

x'  +  Ai 

+  A 

+  A, 

+  A, 

--^o 

-^1 

-A, 

X^+...        (2) 


In  the  first  member  of  (2),  the  coefficient  of  any  power  of  x 
that  does  not  appear  is  zero.  Equating  the  coefficients  of  the 
like  powers  of  x  in  the  two  members  of  (2),  we  obtain 


^0=1,^0  +  ^1=0,  ^2  +  ^i-^o=-i»  ^3  +  ^2-^1  =  0/ 

y^4  +  y^3-^2  =r  0,  ...  ,  A„  +  An-l-An_2  =  ^'. 

Solving  the  system  of  equations  (3),  we  obtain 


A, 


,  A^  —  I,  Aq  _  —  2,  A^ 
,A„  =  A„_2-A„_i. 


As  =  3, 


Substituting  these  values  o(  Aq,  A^,  .  > .  ,  in  (i),  we  have 

I   -X' 


I  +x 


=  1  -x  +  x' 


2x^  +  :^x*  + 


(3) 


(4) 


(5) 


which  is  an  identity  for  such  values  of  x  as  render  the  series 
convergent. 

The  values  of  Aq,  A^,  A^,  "-,  A„,  given  in  (4),  render  (2) 
an  identity;  for  they  satisfy  equations  (3),  and  therefore  render 
the  coefficients  of  like  powers  of  x  in  (2)  identical.     Now  if 


DEVELOPMENT   OF  FUNCTIONS   IN   SERIES.       1 53 


(2)  is  an  identity,  then  (i),  or  (5),  also  is  an  identity  for  such 
values  of  ;ras  render  the  series  convergent  (§  259). 
The  law  of  coefficients  of  the  series  in  (5)  is 

A„^A„_^-A„_^.  (6) 

By  (6)  the  series  can  be  readily  extended  to  any  number  of 
terms.     Thus, 

^6  =  /?8  -  /^4=  -  2  -  3  =  -  5,  ^6  =  ^4-^6  =  3  +  5  =  8, . . . 

I 


Example  2.   Develop 


:^-x^ 


Assume 


^A^x-'^^A^  x-^  +  A^  +  A^x  + 


x^-x^-x* 

Clearing  (i)  of  fractions,  we  have 


1=  Aq  +  A^ 
-A, 


x  +  A.^ 
-A, 
-A. 


x^  +  A, 
-A, 
-A, 


x^  +  A, 
-A, 

-Ao 


x*  + 


(0 

(2) 


Equating  the  coefficients  of  like  powers  of  x  in  (2),  we  have 


Aq  —  i»  A I  —  A, 


0-0,  A^-Aj^-A^  =  0,  A^-A^-A^  =  0^] 


A^-A^~A.,  =  0,...,A„-  A„_i  -  A„_,  =  0.]}^^ 
'.    ^0=  I,  ^1=  I,  ^2  =  2,  ^8  =  3,  ^4=5»-"        (4) 


Here  the  taw  of  coefficients  is  A„  =  A„_i  4-  A„^2. 
Substituting  in  (i)  the  values  in  (4),  we  obtain 

_^_i^-- =^-2  +  ^-1 +  2  + 3^+5^24...., 


(5) 


which  is  an  identity  for  such  values  of  x  as  render  the  series 
convergent.     [Let  the  student  give  the  proof.] 

Note.  The  form  assumed  for  the  series  must  in  each  case 
be  such  that  when  the  equality  is  cleared  of  fractions,  no  power 
of  X  will  appear  in  the  first  member  which  is  not  also  found  in 


1 54  ALGEBRA. 

the  second.  For  otherwise,  the  system  of  equations  obtained 
by  equating  the  coefficients  of  the  hke  powers  of  ;r  will  be  im- 
possible.    For  example,  let  us  assume 


x^  —  x^  —  x^  ^ 

Clearing  of  fractions,  and  equating  the  coefficients  of  x^,  we 
obtain  the  absurdity  i  =  0,  which  shows  that  we  have  assumed 
an  impossible  form  for  the  development.  By  the  laws  of  expo- 
nents in  division  we  know  that  the  first  term  of  the  series  will 
contain  x~^  \  hence  we  assume  the  form  in  (i). 

EXERCISE  25. 

Develop  the  following  functions  by  the  principle  of  unde- 
termined coefficients,  and  verify  the  results  by  division  : 

1  -\-  X                                          x^  -{■  x^  -\-  -i 
^' y V  4-    T : — z 


2. 


I   +   2  X  —  2>^^ 

I  —  3  ^2 

X^  -\-    2  Jf ^  +   t 


6. 


I  +  ^ 

+ 

X' 

I  + 

X 

2  x:^  + 

3 

x^ 

1+^^ 

-^       i~  X  -Y  x^  x^  -{■  2>^^ 

Resolution  of  Fractions  into  Partial 
Fractions. 

264.  In  elementary  Algebra  a  group  of  fractions 
connected  by  the  signs  +  and  —  are  often  united  into 
a  single  fraction  whose  denominator  is  the  lowest 
common  denominator  of  the  given  fractions  (§  64). 


DEVELOPMENT  OF  FUNCTIONS   IN   SERIES.       1 55 

The  converse  problem  of  separating  a  rational  frac- 
tion into  a  group  of  simpler,  ox  partial,  real  fractions 
frequently  occurs.  The  denominators  of  these  par- 
tial fractions  must  evidently  be  the  real  factors  of  the 
denominator  of  the  given  fraction.  These  real  factors 
may  be 

I.  Linear  and  unequal. 

II.  Linear  and  some  of  them  equal. 

III.  Quadratic  and  unequal. 

IV.  Quadratic  and  some  of  them  equal. 

To  present  the  subject  as  clearly  as  possible,  we 
shall  consider  these  cases  separately. 

265.    Case  I.     To  a  lijiear  factor  of  the  dejtomina- 
tor,  as  X  —  a,  there  corresponds  a  partial  fraction  of  the 

form  • 

X  —  a 

Example.     Resolve ^—^ —  into  partial  fractions. 

x^  +  X*  —  2X 

.  2X+ 3  A   ,      B      ,      C  .  . 

Assume  -— — ^- — -  =  —  +  +  — ■ —  .  (i) 

X {x  —  I)  (x  -j-  2)       X       X—  I       x+  2 

Clearing  (i)  of  fractions,  we  have 

2X+  3  =  A  (x-i)  (x+2)  +  B(x+  2)x  +  C(x-  i)x     (2) 
=  (A  +  B+ C)x^+ (A  -^2B- C)x-2A.       (3) 

Equating  the  coefficients  of  like  powers  of  :r  in  (3),  we  have 

A+B+C=0,  A  +  2B-C=2,  -2A  =  3.    •  (4) 


156  ALGEBRA. 

Solving  equations  (4),  we  find 

A  =  -l£  =  i,   C=-i.  (5) 

Substituting  these  values  in  (i),  we  obtain 

--^^:±i_  =  -J-  +  _5 ^ (6) 

x^  +  x^  —  2X  2x      3(;^r—  l)       6(;f+2)* 

The  values  o{  A,  B,  and  C  given  in  (5),  render  (3)  and  there- 
fore (i)  an  identity;  for  they  satisfy  (4),  and  therefore  render 
the  coefficients  of  like  powers  of  x  in  (3)  identical ;   hence  (6) 
is  an  identity. 

If  we  assume  that  (2)  is  an  identity,  the  values  of  Ay  B.  and 
C  may  be  obtained  as  follows  : 

Making  x  =      0,      (2)  becomes       $  =  —  2  A  ;  .♦.  A  =  —  ^. 
Making   x=      i,      (2)  becomes       5  =  3  ^y   .-,  B=      f. 
Making  x  =  —  2,     (2)  becomes  —  i  =  6  Cy    .-.   C  =  —  ^. 

266.  Case  II.  To  r  e^iml  linear  factors  of  the  de- 
nominator as  (x  —  b)',  there  corresponds  a  series  of  r 
partial  fractions  of  the  form 

{x  -  by  ^  (jc  -  by-'  ^      ^  x-b 

Example.     Resolve  7 t^t — \ — ^  into  partial  fractions. 

i^x  —  i)^  (;r  +  I)  ^ 

Assume 

T  A  B  C  ^ 

+  :;73T  +  :;ri-r-        (0 


{x  _  1)2  (;f  4.  I)       (;r  -  1)2  '  jr  -  I    '  ;ir 
Clearing  (i)  of  fractions,  we  have 

\=A{x^-\)-^B{x-\){x-V\)-\-C{x~\y- 
=  iB  +  C)  x^  +  {A  -  2  C)  X  +  A  -  B  +  C.  (2) 


DEVELOPMENT   OF   FUNCTIONS   IN   SERIES.  1 5/ 

Equating  the  coefficients  of  like  powers  of  x,  we  have 

B+C=0,  A-2C=0,  A-B-i-C^i.  (3) 

Hence,                              A  =  :^,  B  =  -  \,  C  =  \.  (4) 


Substituting  these  values  in  (i),  we  have 
I  I  I 


(5) 


(x-iy{x+  I)-  2(;r-i)«      4(1--^)      4(-^+i) 

Equality  (5)  is  an  identity;  for  the  vahies  o(  A,  B,  and  C 
given  in  (4)  satisfy  (3),  and  hence  render  (2)  and  therefore  (i) 
an  identity. 

267.  Case  III.  To  anjy  quadratic  factor  of  the  de- 
nomhiator,  as  x^  +  p  x  +  q,  there  corresponds  a  par- 

.      r    .  .  .      r        .  Ax   +    B 

tial  fraction  of  the  form  -^—, n 

•^  -'         -^  x^4-px  +  q 

x^ 
Example.     Resolve  -^— — ^Z^  into  partial  fractions. 

Assume 

£2 Ax^B         C  D 

(jr2+2)(;r+  i)(;i'- I)  ~    jt*  +  2    "^  :r  +  i  "^  ;r  -  i*       ^^^ 

Clearing  (i)  of  fractions,  we  obtain 

;i-2  =  (^;r  +  ^)(jr2-i)+C(;f2+2)(;r-i)+Z^(;r2+2)(;r+i) 
=  {A^C-VDy^{p-C^B)x'^^(:2.C-\-iD-A)x^2D-zC-B.  (2) 

Equating  coefficients  of  like  powers  of  ;r  in  (2),  we  have 


(3) 
2C+2Z>-/?=0,   2Z>-2C-i?  =  0.  '  ^^^ 

Whence,        /^  =  0,  ^  =  f,  C=  -  ^  Z?  =  J.  (4) 

Substituting  these  values  in  (i),  we  have 

X"-  _  Z  I  T 

j^+2;r-2-3(:r2+2)       6(;ir+ I) +6(;ir- I)*        ^^^ 

(5)  is  an  identity  for  the  same  reason  as  that  given  above. 


158  ALGEBRA. 

268.  Case  IV.  To  x  equal  quadratic  factors  of  the 
denominator,  as  (x^  +  p  x  +  q)'^,  tJiere  corresponds  r 
partial  fractions  of  the  form 

Ax  +  B  Cx  +  r>  Lx-\-M 


{x^^px-\-qY        {x'^ -\- p  X  ^  qy-^  x^-\-px-^q 

In  any  example  under  this  case,  by  clearing  the 
assumed  equation  of  fractions  and  equating  the  co- 
efficients of  like  powers  of  ;r,  we  would,  as  in  the  first 
three  cases,  evidently  obtain  as  many  simple  equa- 
tions as  there  are  undetermined  quantities,  and  the 
values  of  A,  B,  C,  ..-,  M,  thus  determined  would 
make  the  assumed  equality  an  identity. 

269.  In  what  precedes,  the  numerator  is  supposed 
to  be  of  a  lower  degree  than  the  denominator. 
If  this  is  not  the  case,  the  fraction  can  be  sepa- 
rated by  division  into  an  entire  part  and  a  frac- 
tion whose  numerator  is  of  a  lower  degree  than  its 
denom'nator. 

For  example : 

x^  _  5  ^'  -  4 

X^ -\-  1  X'^  -  X  ~  2.^  ^        ^'^  X^+2X^-X-2' 


A  5  ^'-4         ^        ^6       _         I  „_i 

^^^         X^-h2X^~X-2~  3(x-^2)        2  (;f  +  I)  "^  6  (^  -  I) 


X*  ,       t6  I  I 


•    ;r3  +  2;r2-:r-2-'^     ^"^3(r+2)       2(.r+i)       6(j--i) 


DEVELOPMENT  OF  FUNCTIONS   IN   SERIES.       1 59 


EXERCISE  26. 

Resolve  the  following  fractions  into  partial  fractions 

x^-2  3  ^^  -  7  a:  +  6 

7-         (^-i)a 


X—  X" 


X^   I  «                  X"  -V   2X 

:r  +  3  23  ;c  —  1 1  ^'^ 

^*  i^*  +  "x  -  2  *  ^*   (2  ^  -  i)  (9  -  ^O 

^  +3^+2^'  i8jc^+  12  .y  — 3 

4*    (r=-  2  :r)  (i  -  ^^)  *  '''•          (3^+2)« 

2  JC''— 12^**  — SjC+  2  42  —  \^X 

5-          :^^-5^^  +  4  "•    (^+'0^-4)* 


6.    . :9__. 


X^  -\-  X  —  \ 


x^  +  x-i      Ax-\-B      Cx  +  D 

Assume  -^-T^:^  =^  (P^T^^  +  ^^^^TV  * 

2:r*—  iiJf+5  x^  —  2  X  -\-  ^ 

^3-    (^2_^2^_3)(^_3)-     H-    -^2-^77)2- 


*  270.    Reversion  of  Series.      Given 

y^=zax-{-bx'^-\-cx^-{--"f  (l) 

the  series  being  convergent,  to  express  x  in  terms  of 
y  ;  that  is,  to  revert  the  series. 

Assume       a:  =  ^^  +  ^/ +  C/ +  •••  (2) 


i6o 


ALGEBRA. 


Substituting  in  (2)  the  value  ofj/  given  in  (i), 


x^  -\-  Ac 
+  2Bab 


x^  + 


(3) 


Equating  the  coefficients  of  Hke  powers  o{ x  in  (3), 
Aa=^i,  Ab  -\-  B  a^^Q,  Ac+  2Bab  +  Ca^  =  0,  ... 

b        „        2b'' 


Hence     ^  —  -,  B  = 


,    C^ 


ac 


or  a" 

Substituting  these  values  in  (2),  we  obtain 

a  c 


I  b    ^^2b''- 


f- 


which  is  the  result  sought. 

If  the  series  to  be  reverted  be  of  the  form 

y  ^  aQ-\-  ax  ■\-  bx^  -{■  cx^  +  •••, 

we  express  x  in  terms  of  j^  —  a^. 

Example.     Revert  the  series 

y  —  2  +  2X  —  x'^  —  X^+2X^-\---. 


From  (i), 


2  =  2X  —  X^  —  X^+2X^  +  " 


Assume  x=  A  {y  -  2)  +  B  {y  -  2y  +  C  {y  -  2y  + 
Substituting  in  (3)  the  value  oi  y  —  2  given  in  (2), 


x= 2  A  X  —  A 

x^-A 

X^+2A 

+  4B 

.  -4B 

-3B 

+  8C 

-12C 

+  16D 

(0 

(2) 

(3) 


DEVELOPMENT   OF   FUNCTIONS   IN   SERIES.       l6l 

Equaling  coefficients  of  like  powers  of  ;ir,  we  obtain 

2A  =  i,  4B-A=0,  SC-4B-A=0,        ] 
i6Z>-  i2C-3^  +  2^  =  0,  ...J 

Hence,  A=\,  B  =  \,  C  =  \,  Z^  =  A*-- 

Substituting  in  (3),  we  have 

EXERCISE  27. 
Revert  the  following  series  : 

1.  y  =  X  -\-  x^  -\-  x^  -\-  •'» 

2.  y  =  X  —  2x^ -{■  $x^  —  •" 

3.  y  =  x-ix^+  ^X^ 

4.  j=i  +x-hi^+h^'  +  ih^'+  '•' 

5.  y  =  x  +  sx^  +  s^'^  +  7^*  +  "' 

6.  y=2X  +  s^^  +  4^^  +  5^''-^ 

Maclaurin's  Formula. 

271.  Mac/a?/rm's  Formula  is  a  formula  for  develop- 
ing a  function  of  a  single  variable  into  a  series  of 
terms  arranged  according  to  the  ascending  powers 
of  that  variable,  with  constant  coefficients. 


l62  ALGEBRA. 

272.    To  deduce  Maclaurin  s  Formula. 
We  are  to  find  the  values  o{  A^,  A^,  A^,  ...,  when 
/(;r)  can  be  developed  in  the  form 

/{x)  =A,  +  A,x  +  A^x'  +  A,x'  +  A,x'  +  ...     (i) 

in  which  Aq,   A^,  A^,  ••.,  are   constants,  the  series 
being  finite,  or  infinite  and  convergent. 

Finding    the    successive    derivatives    of  (i),    we 
obtain 

/i(x)  =  A,  +  2A,x  +  sA,x''+  4A,x^  +  ...  (2) 

/"  {x):=2A,  +  2.s  A,x  +  3  .4A,x'  +  ...  (3) 

/"'{x)  =  2.sA,+  2.S'4A,x  +  ...         ^  (4) 

/-(^)  =  2.3.4^4  +  -  (5) 

Let  x=0',   then  from   equations   (i)   to  (5),  we 
obtain 

/(O)  =  A„         />  (0)  =  A, ,         /"  (0)  =  2A„ 

/"'(0)  =  \3A„    /-(0)  =  [4^.,      ... 

Solving  these  equations  for  ^0.  ^1  >  ^2,  •••,  we  have 

/"  (0) 


A„  =/(0),         A,  =/'  (0),  A,  = 


11     ' 


A  _/"'(0)      .  _/"(0)      .     .      _/'-'(Q) 

Substituting  these  values  in  (i),  we  obtain 

f{x)^A^)^f\^)x\f\^f^^^  (6) 

which  is  the  required  formula. 


DEVELOPMENT   OF  FUNCTIONS   IN   SERIES.       1 63 

This  formula,  though  bearing  the  name  of  Mac- 
laurin,  was  first  discovered  by  James  Stirling  in  the 
early  part  of  the  last  century. 

The  Binomial  Theorem,  Logarithmic  Series,  Ex- 
ponential Series,  and  many  other  formulas,  are  but 
particular  cases  of  this  more  general  formula. 

Binomial  Theorem. 

273.  The  Binomial  Theorem  is  a  formula  by  which 
a  binomial  with  any  exponent  may  be  expanded  in  a 
series.  Its  general  demonstration  was  first  given  by 
Sir  Isaac  Newton.  It  was  considered  one  of  the  finest 
of  his  discoveries,  and  was  engraved  on  his  tomb. 

274.  To  deduce  the  Binomial  Theorem, 

To  do  this  we  develop  {a  +  jr)'"  by  the  formula 

f{x)  =/(0)+/'(0).r+/"(0)^+...+/"-H^)£^V...    (i) 

Here     f{x)  =  (^  +  .r)-  ;  .-.    /(O)  =  a"^, 

fix)  =  m{a^  or)'"  -^ ;        .-.  /'(O)  =  m  rz'""'. 
f"(x)  =  m{m-i){a  +  x)—^; 

.'.   f"{0)  =  m{m-i)a"^-\ 
f"(,\)  =  m  {m  —  1)  {m  —  2)  {a  +  x)'"-^ ;     ' 

.-.  /'<'{{))  =m(m-i)  (m  -  2)  a"'-\ 

/"-i  (x)  =  m  (m  —  j)...(m~n+  2)  (a  +  x)""  -« ^  1 ; 
•••  /'"'  (0)  =  m  (m  -  i)  ...  (m  -  «  +  2)  a'"--  +  \ 


1 64  ALGEBRA. 

Substituting  these  values  in  formula  (i),  we  have 

7nim~\)        „    „    m(m—i)bn~2) 

LI  lA 

+  . . .  4- — «"*  -  "  + 1 ;»;"  - 1  +  . . . , 


in  which  the  last  term  is  the  /^th,  or  general,  term  of 
the  formula. 

Example.    Find  the  6th  term  in  the  expansion  of  (.r^— ^^)~^. 

Here  n  =  6,  a  =  x^^  x  —  —b'^,  and  w  =  —  | ;  hence  ;;/  —  «  +  2 
=  —  \^.  Substituting  these  values  in  the  «th  term  of  the  for- 
mula, we  obtain 

6thterm  =  ^^lHzlK=lK^V)(z^(,.)-|-5(_,i)^ 

1.2.3.4.5  '       ^       ^  ^       "^    ^ 

729 

275.  By  an  inspection  of  the  Binomial  Theorem  we 
discover  the  following  laws  of  exponents  and  coeffi- 
cients, which  are  very  useful  in  its  applications: 

(i.)  The  exponent  of  a  in  the  first  term  of  the  series 
is  the  same  as  that  of  the  binomial,  and  it 
decreases  by  unity  in  each  succeedifig  tei^m. 

(ii.)  The  exponent  of  x  is  unity  in  the  second  term, 
and  increases  by  unity  in  each  succeeding 
term. 

(iii.)  The  coefficient  of  the  first  term  is  imity,  and  that 
of  the  second  is  the  expottent  of  the  binomial. 


DEVELOPMENT   OF  FUNCTIONS   IN   SERIES.       1 65 

(iv.)  If  i?t  any  term  the  coefficie^it  be  multiplied 
by  the  exponent  of  a,  a7id  this  product  be 
divided  by  the  exponent  of  x  increased 
by  unity,  the  result  will  be  the  coefficietit 
of  the  next  term. 

Example  i.    Expand  (^+3^)*- 
Here  a-c,  x=3y,  m  =  4; 

=  c*  +  12  ^V  +  54  ^V  +  108  cy^  +  Siy*.  (2) 

In  the  series  in  (i)  the  exponent  of  c  in  the  5th  term  is  0; 
hence  by  (iv.)  the  6th  term  is  0,  and  therefore  the  expansion 
consists  of  5  terms. 

Example  2.     Expand  («2  _  c'^)  "i   or  [«2  +  (_  c^)y^. 
Applying  the  laws  given   above,  noting  that  here  a  —  n\ 
x  =  —  c%  and  m  =  —^,  we  have 

=  n~^  +  ^n~^c^  +  ifr'^c*  +  ^{n~^c^  +  ".  (2) 

By(iv.)  the  coefficient  of  the  3d  term  in  (i)  is  (—  ^)  (— |) 
-T-  2,  or  f ;  that  of  the  4th  term  is  f  (-  f )  -^  3,  or  —  |*,  etc. 

In  (i)  the  2d  term  has  two  negative  factors;  the  3d,  two; 
the  4th,  four,  etc.;  hence  the  signs  of  all  the  terms  in  (2)  are  -f . 

This  development  could  be  obtained  by  substituting  ;/2,  —  f2, 
and  —  \,  respectively,  for  a,  x,  and  w,  in  the  formula,  but  the 
process  would  be  longer. 

In  the  series  in  (i),  the  exponent  of  w'^  cannot  be  0  in  any 
term  ;  hence  no  term  can  have  0  as  a  factor  of  its  coefficient, 
and  thus  vanish.  Therefore  the  expansion  is  an  infinite  series, 
and  equals  the  function  only  when  convergent. 


1 66  ALGEBRA. 

276.  When  m  is  a  positive  whole  nnmber,  the  bino- 
mial series  is  finite  and  consists  ^  m  +  i  terms ;  when 
m  is  fractional  or  7iegative,  the  series  is  infinite. 

For  when  m  is  a  positive  whole  number,  the  expo- 
nent of  a  in  the  {^n  +  i)th  term  is  0;  hence  by  (iv.) 
of  §  275  the  {7n  +  2)  th  term  and  all  succeeding  terms 
are  0.     Therefore  the  series  consists  oi  m  ^-  i  terms. 

But  when  m  is  fractional  or  negative,  the  exponent 
of  a  cannot  be  0  in  any  term ;  hence  no  term  can 
have  0  as  a  factor,  and  the  series  is  infinite. 

Thus,  the  expansion  of  {x^yy^  is  a  finite  series  of  14  terms; 
while  the  expansion  of  (:r  4-j)'^  or  of  {x  ^ y)-"^  is  an  infinite 
series. 

277.  When  m  is  a  positive  zvhole  nnmher,  the  coefii- 
cients  of  terms  equidistant  from  the  begimiing  and  e7td 
of  the  expansion  ^  (a  +  x)'"  are  equal. 

For 

m{in—\)        „    „  W 

{a+ xy  =  a""  +  m  a"^-^  x+^^ — a»'-^x^+...  +  T^x'»,      (i) 

{x+  ay  ^x'^  +  m x*"-^ a  +  ^^ — ' x'^'-'^aP'^ '"'^f^^"''       ^^^ 

Now  the  series  in  (2)  has  the  same  terms  as  the 
series  in  (i),  but  in  reverse  order;  whence  the 
proposition.  Hence,  in  expanding  any  positive 
power  of  a  binomial,  after  we  have  computed  the 
coeflficients  of  the  first  half  of  the  series,  the  remain- 
ing coefficients  are  known  to  be  those  already  found 
written  in  reverse  order. 


II 


12 


DEVELOPMENT   OF  FUNCTIONS   IN   SERIES.       1 6/ 

EXERCISE  28. 

1.  If  m  is  a  positive  integer,  what  is  the  sign  of  the  even 
terms  in  {a  —  xY  ?     Why  ? 

2.  Write  out  the  expansion  of  (i  +  a:)'". 

Expand 

ID.    {\-\-df, 

13.   {a~^-2b''c^. 

14.  (i  —  AT^y. 

24.  (9  +  2  jc)^. 

25.  (4^-8^)-*. 

26.    (r^«-2  _^2^-§)-i 

I 
27. 

28. 


6- 

4- 

5- 

(^  + 

by. 

6. 

(2- 

ixy. 

7- 

{r-^ 

-**r. 

8. 

{r^- 

3«-^r. 

9. 

(r 

2x)  • 

Expand  to  five  terms 

15- 

(3  + 

x>)i. 

16. 

(8  + 

12  «)^. 

17- 

{y  + 

.)-». 

18. 

0  + 

a:^)-^ 

19. 

G- 

3-)*. 

20. 

G- 

3-)-4. 

21. 

(«^  + 

.i)?. 

22. 

(- 

d^)-K 

23- 

(/i- 

-c-i)-i. 

a/x" 

-f 

I 

ai- 

-i-h 

a 

'^-  (,^-f_^,v-^^)^ 


1 68  ALGEBRA. 

Find 

30.  The  4th  term  of  (x  —  5)^^. 

31.  The  loth  term  of  (i  —  2  x)'^. 

(  by 

32.  The  5th  term  of  \2  a J . 

II.   The  7th  term  of  f  — ^^-j  . 

34.  The  6th  term  of  (^^  -  c'x')-^. 

35.  The  5th  term  of  (^"^  +  e~^)~^, 

36.  The  7th  term  of  {a^  —  b-^~^^. 

37.  The  6thterm  of  (^~^  — dt^^*)"^ 

278.   To  find  the  ratio  of  the  (n  ■\-\)th  term  to  the  nth. 

Substituting  n  -\-  \  for  n  in  the  ?/th  term  of  the  bi- 
nomial theorem,  we  obtain  as  the  (//  +  i)th  term, 

m{m-  i)(m-2)...im-n+  i)  ^^_„^« 

Dividing  this  by  the  nth.  term,  we  obtain 

(m  —  n  +  i\x          fm  +  I        \  x  ,  >. 
^— )-,  or     --^^ I    -,           (i) 
n         J  a          \    71             )  a 

as  the  ratio  sought;  that  is,  (i)  is  the  quantity  by 
which  we  multiply  the  n\\\  term  to  obtain  the  next 
term. 


DEVELOPMENT   OF  FUNCTIONS    IN   SERIES.       169 

This  ratio  affords  the  following  simple  proof  of  the 
principle  in  §  276: 

When  m  is  a  positive  integer,  this  ratio  is  evidently- 
zero,  for  it  —  ni-\-  I  ;  hence  the  (;//  +  2)th  term  and 
all  the  succeeding  terms  are  zero,  and  therefore  the 
series  consists  of  ;«  +  i   terms. 

But  when  ;«  is  fractional  or  negative,  no  value  of  n 
(ft  must  be  integral)  will  make  the  ratio  zero ;  hence 
no  term  can  become  zero,  and  the  series  is  infinite. 

279.  Any  root  of  a  number  may  be  found  approx- 
imately by  the  Binomial  Theorem. 

Example.     Find  the  approximate  5th  root  of  248. 

^^  =  (243  +  5)^ 

=  (3'  +  5)* 

(I  22.3  \ 

=  3(1+  0.0041  152  -  0.0000338  +  0.0000004  -  •  •  •) 
=  3.0122454, 

which  is  correct  to  at  least  six  places  of  decimals. 

280.  Expressions  which  contain  more  than  two 
terms  may  be  expanded  by  the  Binomial  Theorem. 

Example.     Find  the  expansion  of  {x^  +  2  or  —  i)*. 
Regarding  -zx  —  \  as  a  single  term,  we  have 
[;r2  +  (2:r- i)P  =  (.1-2)8  + 3(:r2)2(2;r- i)  +  3a-2(2;t—i)2+(2;r- 1)8 
=  x^  +  6x^  +  ()x^—^x^—^x'^+6x—i. 


I/O  ALGEBRA. 

EXERCISE  29. 

Expand  and  write  the  «th  term  of 

I.    (i_^)-i.  2.    {i-x)-\  3.    {i-x)-\ 

Find  to  five  places  of  decimals  the  value  of 

4-    V'ai-  6.    v^^.  8.    ^^2400. 

Find  the  expansion  of 

10.    (1  +  2^  —  ^^2)4.  II.    {^x'^—  2ax  -^  :^a'^Y. 

Find  the  /zth  term  of  the  expansion  of 

4 


14. 


3(1  —  2^)  (2  —  ^)^ 

5  _         3:^2  +  ^—2 


^^'    3(2-^)*  '^'    (^-2)2(1-2:^)' 

By§266,-l£^±-£ll^  =  _ ^ +-5 -1- 

•^  ^        '(;r-2)2(r-2;r)  3(i-2;r)      3(2-;r)     {2-xf 

Hence,  by  Examples  12,  13,  and  14,  the  «th  term  is 

V  3  3      2"        2"-V 


I  +  llJC+28^=^  ''    (i  —  Jt:^)  (i  —  2a:) 

Expand  to  four  terms  in  ascending  powers  of  x 

(2  +  ;,)  (i  _  :r)  •  ^-    (x  -  I)  (^2  +  i)  * 


CONVERGENCY   OF   SERIES.  I71 


CHAPTER   XV. 

CONVERGENCY    AND     SUMMATION   OF    SERIES. 

281.  An  infinite  series  is  divergent,  if  the  nth  term 
does  not  approach  zero  as  its  limit  when  «  =  x.  For 
if  the  nth  term  does  not  approach  zero  when  ;/  =  x, 
the  sum  of  n  terms  cannot  approach  a  hmit. 

Thus,  the  series  -_:^_^4_5^._  ^  'i±_L  ^  . . .  js  diver- 
1234  « 

gent ;   for  the  «th  term  approaches  unity  and  not  zero  as  its 

limit. 

A  series  may  be  divergent  even  though  the  «th 
term  approaches  zero  as  its  limit  when  7i  =  x. 

Example.     Show  that  the  harmonic  series 

I  +  -  +  -H [-...-f_  +  ...  is  di  versrent. 

234  n 

If  after  the  first  two,  the  terms  of  this  series  be  taken  in 
groups  of  two,  four,  eight,  sixteen,  etc.,  we  have 

Each  parenthetical  expression  is  evidently  greater  than  ^. 
Regarding  these  as  single  terms  of  series  (i),  the  sum  of  7n 
terms  is  greater  than  ^m.  But  w  increases  indefinitely  with  n. 
Hence  the  series  is  divergent,  although  its  «th  term  =  0,  when 


172  ALGEBRA. 

282.  The  following  three  important  principles  are 
almost  self-evident : 

(i.)    An  infinite  series  of  positive  terms  is  conver- 
gent, if  the  sum  of  its  first  n  terms  is  always 
less  than  some  finite  quantity,  however  large 
n  may  be. 
For  as  the  sum  of  71  terms  must  always  increase 
with  fly  but  cannot  exceed  a  finite  value,  it  must  ap- 
proach some  finite  limit. 

(ii.)  If  a  series  in  which  all  the  terms  are  positive 
is  convergent,  then  the  series  is  convergent 
when  some  or  all  of  the  terms  are  negative, 
(iii.)  If,  after  removing  a  finite  number  of  its  terms, 
a  series  is  convergent,  the  entire  series  is 
convergent;  if  divergent,  the  entire  series  is 
divergent.  For  the  sum  of  this  finite  num- 
ber of  terms  is  finite. 

283.  An  infinite  series  ift  which  tJie  terms  are  alter- 
nately positive  and  negative  is  convergent,  if  its  teiiris 
decrease  numerically,  and  the  limit  of  its  nth  terin  is 
zero. 

Let  the  terms  of  the  series  be  denoted  by  n^,  —  n^, 
u.^y  .  .  .  y  and  their  sum  by  s  ;  then 

s  —  u^  —  u^Ar  u^  —  u^-\-  u^ .  ±  u„  :f  ...     (i) 

Since  u„  =  0  when  ;/  =  x,  the  sum  of  the  series  is 
evidently  the  same  whether  we  take  an  even  or  an 
odd  number  of  terms. 


CONVERGENCY   OF  SERIES.  1 73 

Now  (i)  may  be  written  in  the  form 

s  =  ii^  —  («2  —  u^)  —  (//^  —  u^ (2) 

or        s  =  (//,  -  //2  )  +  ('^3  -  '0  +  ('^5  -  ^O  +  •  •  •  (3) 

Since  «i  >  //^  >  «3  >  •••  >  ^«>  the  expressions 
u^  —  «2>  ^2  —  ^'sj  ^3  —  "4>  •  •  •  2ire  all  positive.  Hence 
from  (2)  we  know  that  s  <  u^;  therefore  by  (i.)  of 
§  282  the  series  in  (3)  is  convergent. 


lyi 


Thus,  the  series  ^    humrvM^  ^ 

I  I  I  I       I     \     ^  V' 

I 1 1 ±-T..«    IS  convergent. 

2^3      4      5  « 

If  we  put  this  series  in  the  forms 

-e-^e-;)---'(-i)*(i-^ 

we  see  that  its  sum  is  less  than  i  and  greater  than  ^. 


284.  Ah  infinite  series  is  convergent  if  the  ratio  of 
each  term  to  the  preccdirig  (erni  is  less  tJian  some  fixed 
quantity  that  is  itself  numerically  less  than  unity. 

Let  all  the  terms  be  positive ;  then 

f  =  Wj  +    //g   +    //g   +    ^'4  H 

=  „.f,  +  '^  +  «»  +  «.  +  ...) 

\  U^         «1  «i  / 


(rd- 


174  ALGEBRA. 


Let  ^  be  a  fixed  quantity  less  than   i,  but  greater 
than  any  of  the  ratios    —  >  -  ^  —  >  •  •  • ;  then  from  (i) 


5  <  U,{y   +  ^  +  ^2  _^  ^3  _^  .  .  .)^ 

or        •$■  <  2^1 T  >    a  finite  quantity.  §  259,  Ex.  2. 

Hence  by  (i.)  and  (ii.)  of  §  282  the  series  is  conver- 
gent whether  its  terms  are  all  positive  or  some  or  all 
negative. 

285.  An  infinite  series  is  divergent  if  the  ratio  of 
each  term  to  the  preceding  term  is  numerically  equal 
to  or  greater  than  unity. 

For  if  this  ratio  is  unity,  or  greater  than  unity,  the 
/2th  term  cannot  approach  zero  as  its  limit,  and  the 
series  is  divergent  by  §  281. 

286.  In  the  application  of  the  tests  of  §  §  284,  285, 

it  is  convenient  to  find  -^^'    ;   let  this  limit  be 

n  —  y^V  u^  J 

denoted  by  r. 

If  r  <  I,  the  series  is  convergent.  §  284. 

If  r  >  I,  the  series  is  divergent.  §  285. 

\i  r  —  I,  and  u^^^~  u^  >  i,  the  series  is  divergent 
by  §  285  ;  if  r  =  T,  and  u„^^  ~-  u„  <  i,  the  test  of  §  284 
fails,  and  other  tests  must  be  applied. 


CONVERGENCY  OF   SERIES.  1 75 

Example  i.     For  what  values  of;iris  the  logarithmic  series 
x^      x^      X*  X"      jr"  +  i 

convergent? 

Here     ''-*   M  =    '™''  [f     '      _  ,U  =  _;.. 

Hence,  if  :r  <  i  numerically,  the  series  is  convergent. 
If  jr  >  I  numerically,  the  series  is  divergent. 
If  X  =  I,  the  series  is  convergent  by  §  283. 
If  jr  =  —  I,  the  series  becomes   —  (i  +  ^  +  J  +  •••)» 
and  is  divergent  by  Example  of  §  281. 

Hence  (i)  is  convergent  for  jr  =  i,  or  .r  >  —  i  and  <  -f  I. 

Example  2.     When  the  binomial  series  is  infinite,  for  what 
values  of  x  is  it  convergent? 

Here     '™"  [^^'1  =    '™"  r(^^'_Afl  =  ^-  §278. 

Hence,  ii  x  <ia  numerically,  the  series  is  convergent. 
If  ;r  >  rt  numerically,  the  series  is  divergent. 
U  x  =  a  numerically,  the  test  of  §  284  fails. 

Thus,  the  theorem  will  develop  (8  +  2)2,  but  not  (2  +  8)2, 
Hence  when  m  is  fractional  or  negative,  the  binomial  theo- 
rem will  give  the  development  of  (a  +  x)"*  or  (x  4-  «)'",  ac- 
cording as    ;r  <  or  >  a  numerically.      li  x  =  a    numerically, 
{a  +  xy^  becomes  (2^)"'  or  0'",  and  the  formula  is  not  needed. 

Example  3.     For  what  values  of  x  is  the  series 

p+^  +  ^  +  ^.+ •••  +  -.  + •••convergent? 


1 76  ALGEBRA. 

(i.)  If  ;ir>  I,  the  first  term  is  i;  the  sum  of  the  next  two 

2 

terms  is  less  than  — ;  the  sum  of  the  next  four  terms 

4 
is  less  than—;   the  sum  of  the  next  eight  terms  is 

8 
less  than  ■^;  and  so  on.     Hence  the  sum  of  the  series 

2  A  R 

is  less  than  that  of     i-i -l^-l— _|_..., 

2^  ^  4^^  ^  8-^  ^       ' 

which   is  a  geometrical  progression   whose  common 

ratio,    2  -^  2*',   is   less   than   i  ;   hence   the  series  is 

convergent. 

(ii.)  If  ;ir  =  I,  the  series  is  the  harmonic  series,  and  is  diver- 
gent by  Example  of  §  281. 

(iii.)  If  JT  <  I,  each  term  is  greater  than  in  case  (ii.),  and 
therefore  the  series  is  divergent. 


EXERCISE   30. 

Determine  which  of  the  following  series  is  convergent  and 
which  divergent : 

2'       3'      4' 


i     ^     11 


5-    ■^+-  +  -  +  -  +  - 
234 


SUMMATION   OF   SERIES.  1 7/ 


«-  +  ^^^  +  ^»  + 


1.2       2-3       3-4 


8.    + +  + + 

1-2       2-3       3-4       4.5 


X  X^  X^  X* 

2^  3*  « 


SUMMATION    OF    SERIES. 

287.  The  Summation  of  a  series  is  the  process  o< 
finding  an  expression  for  the  sum  of  its  first  ;/  terms. 

Formulas  for  the  sum  of  the  first  ;/  terms  of  an 
A.  P.  and  of  a  G.  P.  were  obtained  in  Chapter  XI. 
We  proceed  to  deduce  formulas  for  the  sum  of  other 
series. 

Recurring  Series. 

288.  When  the  «th  term  of  the  series 

t^l  +  «2  +  //g  +  7/^  + 1-  u„_,  +  u„ 

is  connected  with  the  m  preceding  terms  by  a  rela- 
tion of  the  form 

the   series    is    called    a  Recurring  Series  of  the   mth 


178  ALGEBRA. 

order.  The  multipliers  pi,  p^,  •  •  ',p^  remain  unchanged 
throughout  the  series. 

In  the  G.  P.  i  +  2;ir  +  ^x'^  +  ^x^  +  •  •  •,  Un  —  ^^'  ^h—l',  hence 
this  series  is  a  recurring  series  of  ihtjirst  order,  in  which  2;r  is 
the  multiplier. 

In  the  series  i  -\-  2 x -\- Z x'^  +  28  r3  +  ioo;ir4  +  ...  (i) 

we  have  Un  —  Z^-'^n-i-\'^^'^''^n-2'  (2) 

Thus  2>x'^  =  ^x.2x+2x^'i. 

and  28  ;r8  =  3  ;ir .  8  y-^  +  2  ^"^  •  2  :r. 

Hence  series  (i)  is  a  recurring  series  of  the  second  order  in 
which  3  X  and  2  jr^  are  the  /w^  multipliers. 

The  series  i  +  3  +  7  +  i3  +  2i  +  3iH is  a  recurring  series 

of  the  third  order,  in  which  the  three  multipliers  are  3,  —3,  and 
I.     Thus  31  =  3  X  21  -  3  X  13  +  7- 

289.  If  we  have  given  the  m  muhipliers  of  a  recur- 
ring series  of  the  mth.  order,  any  term  can  be  found, 
if  we  know  the  m  preceding  terms. 

Thus,  to  find  the  6th  term  of  series  (i)  in  §  288,  we  have 
Mq  —  '^x-  loox^  +  2x'^  ■2^x^  =  356;lr^ 
To  find  the  7th  term  of  the  last  series  in  §  288,  we  have 
7/y  =  3  X31  -3  X  21  +  13  =  43. 

290.  To  find  the  midtipliers  of  a  recurring  series. 

(i.)    If  the  series  is  of  the  first  order,  let  px  be  the 
multipHer,  then 

u^,  =Pi  u^y  or/i  =  u^-^  Uy 


SUMMATION    OF    SERIES.  1/9 

(ii.)    If  the  series  is  of  the  second  order,  let/i  and 
/a  be  the  multipliers ;  then 

and  «/4=A^3+A^2- 

From  these  two  equations  the  values  of  A  and 
p^  may  be  found  when  the  first  four  terms 
of  the  series  are  known. 

(iii.)  If  the  series  is  of  the  third  order,  let  /i,  A* 
and  /a  be  the  multipliers  ;  then  from  any 
six  consecutive  terms  we  can  obtain  three 
equations  which  will  determine  the  values 
of  A'  A»  and  py 
If  the  series  is  of  the  wth  order,  we  must 
have  given  2  m  consecutive  terms  to  find 
the  m  multipliers. 

Example.     Find  the  multipliers  of  the  recurring  series 

Let  the  multipliers  be/j  and /a  ;  then  to  obtain /j  and/21  we 
have  the  equations 

13  ;r2  =  5  xpx  +  2/2  and  35  ^8  =  13  x"^ p^  +  5  xp<i. 
Hence  /i  =  5  ;r  and/2  =  — 6:r2. 

Remark.  In  finding  the  multipliers,  if  we  assume  too  high 
an  order  for  the  series  we  shall  find  one  or  more  of  the  assumed 
multipliers  to  be  zero.  If  too  low  an  order  is  assumed,  the  error 
will  be  discovered  in  attempting  to  apply  to  the  series  the  mul- 
tipliers found. 


l8o  ALGEBRA. 

291.    To  find  the  sum  of  a  recurring  series. 

If  the  series  is  of  the  first  order,  see  §  211. 

If  the  series  is  of  the  second  order,  let/^  and /^  de- 
note the  multipliers,  and  5„  the  sum  of  n  terms;  then 

Sn  —  Ux-\-U^        -V   U^        H 1-   tin 

—A  Sn  =        —A  U^-pxU., A  Un-^  —  p^  U„ 

—A  S,,  =  — /2  U^ A  ^n-a  —A  ^^n  - 1  —  A  ^^«- 

Adding  these  equalities,  and  noting  that 

^8  —  A  ^2  —  A  ^^1  =   0»   •  •  •»   ««  —A  ^n-^     —ptUn-^^  =  0, 

we  obtain 

^  —  ^1  ('  ~  A)  +  ^h     A  ^«  +  A  (^»-i  +  ti„)      .X 
I— A— A  I— A— A 

If  the  series  is  infinite  and  convergent,  (i)  becomes 


I  -A  -/: 


Similarly,  if  the  series  is  of  the  third  order,  and 
/p  /)j,  and /a  are  the  multipliers,  we  obtain 

<,  _  u,  (i  — /i  —A)  +  ih  (i  —A)  +  ^^3 
I -A -A -A 

A  ^»   +  A  (^^.-i    +    ?^„)   4-  A  (^n-.  +    Un-r   +    ?0        (^^ 

I— A —A— A 
of  which  the  first  fraction  equals  ^^o. 

Unity  minus  the  sum  of  the  m  multipliers  of  a  re- 
curring series  is  often  called  its  Scale  of  Relation. 


SUMMATION   OF   SERIES.  l8l 

292.  From  (2)  and  (3)  of  §  291  we  learn  that  the 
sum  of  a  recurring  series  is  a  fraction  whose  denomi- 
nator is  the  scale  of  relation  of  the  series. 

By  developing  the  fraction  in  (2)  we  could  obtain 
as  many  terms  of  the  original  series  as  we  please ; 
for  this  reason  this  fraction  is  called  the  Generating 
Function  of  the  series. 

EXERCISE  31. 
Find  the  generating  function  of  each  of  the  following  series: 

1.  l   +  2X+^X^  +  4X^+SX*  +  ." 

The  scale  of  relation  is  i  —  2  ;c  +  .v^ :  6" »  =  ; ^  • 

(i  -  xy 

2.  I  +  2X  +  8x^+  28x^  +  lOOX*  +  '^. 

3.  I  +  ^  +  5  -^^  +  13  -^^  +  41  ^*  +  •  •  • 

4.  I  +  SX+  gx^  +  i:^x^  +  *'. 

5.  2  +  3a:+5.v2  +  9a«  +  ... 

6.  I  +  X  +  2  x^  -{-  2  a:^  +  3  Jt'*  +  3  JC^  +  4  a:®  +  4  a:'  +  .  •• 

7.  I  +  ^x  +  6x^  +  iix^  -\-  2Sx*  +  e^x^  +  -'^ 

8.  2  —X  +  2X'^  —  SX^  +   lOX*—  I'JX^  +  ... 

9.  ^  +  6x  +  i4x^  +  ^6x^  +  gS  X*  +  2']6x^  +  ..' 


1 82  ALGEBRA. 

Method  of  Differences. 

293.  If  each  term  of  a  series  be  subtracted  from  its 
succeeding  term,  the  remainders  thus  obtained  form 
the  scries  of  first  differences  ;  the  remainders  obtained 
by  subtracting  each  term  of  this  series  from  its  suc- 
ceeding term  form  the  series  of  second  differences ; 
and  so  on. 

In  an  arithmetical  series,  the  second  differences 
vanish.  In  certain  other  series,  the  third,  the  fourth, 
the  fifth,  or  the  rth  differences  vanish. 

Thus,  if  the  series  is      i,     4,     9,     16,     25,     36,  ... 
1st  differences,  3,     5,     7,       9,     11,  ... 

2d  differences,  2,     2,      2,      2,  ... 

3d  differences,  0,     0,      0,  ... 

Here  the  3d  and  all  succeeding  differences  vanish. 

294.  To  find  the  nth  term  of  the  series 

u^,  u^,  u^,  u^,  u.,  u^,  ... 
Here  the  series  of  successive  differences  are 

1st,      U^  —  U^,     th  —  ^i^     U^  —  U^,     U^  —  U^,'" 

2d,       2^3  —  2  «2  +  Ui  ,    U^—  2U^^r  U^,     U^—  2U^-{-U^,"* 

3d,     n,  —  37/3  +  3  ^^'2  —  ^h '    ^^5  —  3  ^h  +  3  ^-^3  —  '^2'  ••• 

4th,  7/5  —  47/4  +  6  7/3  —  47/2  +  7/1,... 


SUMMATION   OF   SERIES.  I  S3 

Let  Z>i,  /?2»  A»  •••  Dr->  denote  respectively  the 
first  terms  of  the  successive  series  of  differences; 
then 

Z>i  =  u^  —  u^;  .-.  Ui  =  7/1  -f  A • 

Z>2  =  «3  —  2  «2  -r-  ^1 ;         •*•  ^h  —  u^+  2  D^-\-  D^. 

A  =  ^4— 3^^3  +  3 «2 - ^1 ;  •'•  ^4= ^1  +  3  A  +  3 A  +  A • 

A  =  «5  —  4  ?^4  +  6  7/3  —  4  u.-^  +  ?/i ; 

.-.  u,  =  //,  +  4  Z?,  +  6Z>,  +  4  A  +  A- 

The  reader  will  notice  that  the  coefficients  in  the 
value  of  H^  are  those  in  the  expansion  of  (^  +  xy ; 
a  similar  relation  evidently  holds  between  ?/g  and 
{a  +  xYy  21^   and  (a  +  ,i')^,  etc.  ;   hence 

,/  -,.  ^(^   i\n  ,(^-0(«-2)r.  ,  («-i)(«-2)(;y-3) 

Example.     Find  the  loth  term  of  the  series 
I,  2,  6,  15,  31,  56,  ... 

Here  the  successive  series  of  differences  are : 
1st  differences,     i,       4,      9,       16,       25,  ... 
2d    differences,         3,       5,       7,        9,  ••• 
3d    differences,  2,       2,       2,  ... 

4th  differences,  0,      0,  ... 

Hence  ^/j  =1,  «  =  10,  T^j  =1,  A  =  3,  T^g  =  2,  Z>4  =  0. 
Substituting  these  values  in  formula  (i),  we  obtain 
«j^j  =  I  -}-  9  +  108  +  168  =  286. 


1 84  ALGEBRA. 

295.    To  find  the  sum  of  n  terms  of  the  series 

?/j,   ?/.,,   u^,   u^,   ?/,,   u^,  ...  (i) 

Assume  the  new  series 

0,     U^  ,     U^  +   //2  J     ^ti    +  Z^2  +   ^3  »     ^\  +   ^^2  +   ^3  +   «4  »  •  •  •      (2) 

Now  the  sum  of  n  terms  of  series  (i)  is  evidently 
equal  to  the  {11  +  i)th  term  of  series  (2).  Moreover, 
the  series  of  first  differences  of  series  (2)  is  series 
(i)  ;  hence  the  second  differences  of  series  (2)  are 
the  first  differences  of  series  (i);  the  third  differ- 
ences of  series  (2)  are  the  second  differences  of 
series  (i);    and  so  on. 

Hence  we  may  obtain  the  (;/  +  i)th  term  of  series 
(2),  or  S„  of  series  (i),  by  putting  in  (i)  of  §  294 

Making  these  substitutions,  we  have 
Example.     Find  the  sum  of  n  terms  of  the  series 

t2     o2      -j2      a2      c2      ...       «2 

1st  differences,        3,       5,       7,      9,  ••• 
2d    differences,  2,       2,       2,  ... 

3d    differences,  0,      0,  ... 

Hence  u-^  =1,  D^  =  3,  D^  =2,  Z^g  =  0. 
Substituting  these  values  in  the  formula,  we  obtain 

=  J  «  (2  «2  +  3  «  +  I)  =  J  K  («  +  l)  (2  «  +  I). 


SUMMATION  OF  SERIES.  1 85 

EXERCISE  32. 

1.  Find  the  7th  term  of  the  series  3,  5,  8,  12,  17,  ... 

Ans.   30. 

2.  Find  the  15th  term  of  the  series  3,  7,  14,  25,  41,  ... 

3.  Find  the  7th  term  of  the  series  286,  205,  141,  92,  56,  ... 

4.  Find  the  9th  term  of  the  series  194,  191,  174,  146, 
no,   ... 

5.  Find  the  «th  term  of  the  series  i,  3,  6,  10,  15,  21,  ... 
Find  the  sum  of  each  of  the  following  series : 

6.  I,  3,  5.  7,  9>    ••,  2;/-  I. 

7.  2,  4,  6,  8,  ...,  2n, 

8.  I^3^5^7^ -.(^^-i)'. 

9.  2^4^6^8^...,(2«)^ 

10.  m  +  I,  2  (;;/  +  2),  s(m  +  3),  ...,   n(m  +  n). 

Ans.  S„  =  ^  n  (n  +  1)  (3  ///  +  2  //  +  i). 

11.  Find  the  number  of  balls  that  can  be  placed  in  an 
equilateral  triangle  with  «  on  a  side ;  that  is,  find  the  sum  of 
the  series  i,  2,  3,  4,  5,  ...,  n.  Ans.  ^n  (n  +  i). 

12.  Obtain  the  series  whose  nth  term  is  ^n(n  +  i),  and 
find  the  sum  of  n  terms.  Ans.  J  //  (//  +  i)  {n  +  2). 

13.  Show  that 

x«+  2«  +  3^  +  4'  +  •••  +  ^'  =  (i  +  2  +  3  +  4  +  ...  +  n)\ 


1 86  ALGEBRA. 

296.  Application  to  Piles  of  Balls.  An  interesting 
application  of  the  preceding  theory  is  that  of  find- 
ing the  number  of  cannon-balls  in  the  triangular  and 
square  pyramids,  and  rectangular  piles,  in  which  they 
are  placed  in  arsenals  and  navy-yards. 

Triangular  Piles.  When  the  pile  is  in  the  form 
of  a  regular  triangular  pyramid,  the  top  course  con- 
tains one  ball,  the  second  course  contains  three  balls, 
and  the  n\\\  course  from  the  top  is  a  triangle  of  balls 
with  ;^  on  a  side,  and  therefore  contains  \  n  (n  +  i) 
balls  (Example  ii  of  Exercise  32).  Hence  the 
whole  number  of  balls  in  a  triangular  pyramid  hav- 
ing n  balls  on  a  side  of  its  bottom  course  is  the 
sum  of  the  series 

I,  3,  6,  10,  T5,  21,  ...,  ^n(n  +  i). 

Hence  by  Example  12  of  Exercise  32, 

S,,  =  ^n(n+  i){n+2).  (i) 

Square  Piles.  When  the  base  of  the  pile  is  a 
square  having  n  balls  on  a  side,  the  top  course  con- 
tains one  ball,  the  second  course  2^  balls,  the  third 
course  3^  balls,  and  the  ;^th  course  71^  balls.  Hence 
the  number  of  balls  in  the  pile  is  the  sum  of  the 
series 

.    Hence  by  Example  of  §  295 

Sn  =  i  «  («  f  I)  (2  ;^  +  i).       *  (2) 


SUMMATION   OF   SERIES.  1 8/ 

Rectangular  Piles.  When  the  base  of  the  pile  is 
a  rectangle  having  n  balls  on  one  side  and  /«  +  «  on 
the  other,  the  top  course  will  be  a  single  row  of  w  +  i 
balls;  the  second  course  will  contain  2  {in  +  2)  balls; 
the  third  course  3  (;;/  +  3)  balls;  and  the  bottom 
course  n  (m  +  «)  balls. 

Hence  the  number  of  balls  in  the  pile  is  the  sum 
of  the  series 

m  -{■  1,  2(m  -\-  2),  2>{^  -^  3)>  •••>  n(m-\r  n). 

Hence  by  Example  10  of  Exercise  32, 

Sn  =  ln  {n  +1)  (3  fn  +  2  «  +  i).  (3) 

If  we  put  m  =  0,  (3)  becomes  identical  with  (2), 
as  it  should ;  for  when  m  =  0,  the  pile  is  a  square 
pyramid. 

Incomplete  Piles.  If  the  pile  is  incomplete,  find  the 
number  of  balls  in  the  pile  supposed  complete,  then 
find  the  number  in  the  part  that  is  lacking,  and  sub- 
tract the  last  number  from  the  first. 


EXERCISE  33. 

1.  Find  the  number  of  balls  in  a  triangular  pile  of  12 
courses.  How  many  balls  in  the  lowest  course?  How 
many  in  one  of  the  faces? 

2.  If  from  a  triangular  pile  of  20  courses,  8  courses  be 
removed  from  the  top,  how  many  balls  will  be  left? 


1 88  ALGEBRA. 

3.  If  from  a  triangular  pile  of  h  courses,  c  courses  be 
removed  from  the  top,  how  many  balls  will  be  left? 

4.  How  many  balls  in  a  square  pile  of  25  courses?  How 
many  balls  in  each  face  ? 

5.  How  many  balls  in  a  square  pile  having  256  balls  in  its 
lowest  course  ? 

6.  Find  the  number  of  balls  in  the  lower  12  courses  of  a 
square  pile  having  20  balls  on  each  side  of  its  lowest  course. 

7.  The  top  course  of  an  incomplete  triangular  pile  con- 
tains 2 1  balls,  and  the  lowest  course  has  20  balls  on  a  side. 
How  many  balls  in  the  pile  ? 

8.  Find  the  number  of  balls  in  an  oblong  pile  whose  low- 
est course  is  52  balls  in  length  and  21  in  breadth.  If  11 
courses  were  removed  from  the  top  of  this  pile,  how  many 
balls  would  be  left? 

9.  Find  the  number  of  balls  in  an  incomplete  oblong  pile 
whose  top  course  is  10  balls  by  30,  and  whose  bottom 
course  is  45  balls  in  length. 

10.  Find  the  number  of  balls  in  a  rectangular  pile  which 
has  II  balls  in  the  top  row  and  875  in  the  bottom  course. 

297.  Interpolation  is  the  process  of  introducing  be- 
tween the  terms  of  a  series  intermediate  terms  which 
conform  to  the  law  of  the  series.  It  is  used  in  find- 
ing terms  intermediate  between  those  given  in  mathe- 
matical tables,  but  its  most  extensive  application  is 
in  Astronomy. 


SUMMATION   OF   SERIES.  189 

The  formula  for  interpolation  is  that  for  finding  the 
nth.  term  of  the  series  by  the  inetJiod  of  differences. 
Thus  to  find  the  term  equidistant  from  the  ist  and 
2d  terms  of  a  series  we  put  ;/ =  \\  in  (i)  of  §  294; 
to  find  the  term  equidistant  from  the  2d  and  3d  terms 
we  put  n  —  2|. 

Example  i.  Given  log  97=  1.9868,  log 98=  1.9912,  log 99 
=  1-9956;  find  log  97.32. 

Series,  1.9868,  1.9912,  1.9956. 

1st  differences,  0.0044,  0.0044. 

2d  differences,  0. 

Hence     u^  =  1.9868,  Z>i  =  0.0044,  D^  =  0,  u  =  1.32. 
•*•     log  97-32  =  1.9868  +  0.32  X  0.0044 
=  1.9882. 

Example  2.  Given  -v^is  =  3-55^89,  ^47  =  3.60882,  ^49 
=  365930,  v^iT  =  3-70843  ;  find  ^48. 

Here  «i  =  3.55689,  Z>i  =  0.05 193,  /?,  = -0.00145,  Z>j  =  0.0001, 
«  -  I  =  |. 

Hence 

^48  =  3  55689  +  I  (0.05193)  -f  I  (-  0.00145)  -  ^\  (o.oooi) 
=  3-63424. 

EXERCISE  34. 

1.  Given  Vs  =  2.23607,  V6  =  2.44949,  Vj  =  2.64575, 
VS  =  2.82843  ;    find   Vs^,   a/6^. 

2.  Given  the  length  of  a  degree  of  longitude  in  latitude 
41°  ==45.28  miles;  in  latitude  42°  =  44.59  miles;  in  lati- 


I90  ALGEBRA. 

tude  43°   =  43-88  miles;    in  latitude  44°   =  43- 16    miles. 
Find  the  length  of  a  degree  of  longitude  in  latitude  42°  30'. 

Ans.   44.24  miles. 

3.  If  the  amount  of  ^  i  at  7  per  cent  compound  interest 
for  2  years  is  ;^i.i45,  for  3  years  ^1.225,  for  4  years 
^1.311,  and  for  5  years  ^1.403,  what  is  the  amount  for 
4  years  and  9  months  ?  for  3  years  and  6  months  ? 

298.  The  summation  of  some  series  is  readily  ef- 
fected by  writing  the  series  as  the  difference  of  two 
other  series. 

Example  i.    Sum  the  series 


1.2       2.3       3.4       4.5  «(«+!) 

TV  ^       —  '  •  '       _  ^         ^  . 

I   .  2  ~  2'  2.3        2        3  '    *** 

Writing  the  positive  and  negative  terms  separately,  and  de- 
noting the  sum  of  n  terms  of  the  given  series  by  S^  we  have 

[^  \2       3       4  n/      n+  I  ^ 

If  the  series  is  infinite,  (i)  becomes  6'oo  =  i« 

Example  2.   Sum  the  series  + H 2  +  «»» 

1.4     2.5      3.6 

Here  the  «th  term  is  evidently   -- — ; — :  • 


SUMMATION   OF   SERIES.  19I 

Now   --L_  =  '(l--^);  §265. 


S„=' 


3I  _fI  +  ...  +  M ^ ' L_  I 

^ViL ^^ -^ '-). 

3\  6     A/  +  I    «  +  2    //  +  a^' 

Hence  6'»  =  {\. 

Example  3.   Sum  the  series   i  4-  5  +  i  +  ^  +  ••• 

Multiplying  and  dividing  by  2,  we  have 

.'.  »S*»=  2  .  Example  i. 


2 

2« 


«+  I 


EXERCISE   35. 

Find  the  «th  term,  the  sum  of  n  terms,  and  the  sum  of 
all  the  terms  in  each  of  the  following  series : 

1-3       3-5       5-7 

4,4.4,        4        , 

2. 1 1 +  ••• 

1.5       5.9       9  .  13       13-17 


192  ALGEBRA. 


+  -^  +  ^.  +  . 


3-4       4-5       5-6 


'  +-^  + 


2.7       7.12       12-17 


6. 


;^  '  S       6  .  12       9  .  16 
7.   The  series  of  which  the  nth  term  is 


8.    -^  +  -^—  + 


(3'^+2)  (3«  +  8) 


I  •  4      4  •  7       7  •  10 

9.  Sum  «  terms  of  the  series  i,  2*,  3*,  4*,  ... 

10.  Show  that  the  number  of  balls  in  a  square  pile  is  one- 
fourth  the  number  of  balls  in  a  triangular  pile  of  double  the 
number  of  courses. 

11.  If  the  number  of  balls  in  a  triangular  pile  is  to  the 
number  of  balls  in  a  square  pile  of  double  the  number  of 
courses  as  13  to  175,  find  the  number  of  balls  in  each  pile. 

12.  The  number  of  balls  in  a  triangular  pile  is  greater  by 
150  than  half  the  number  of  balls  in  a  square  pile,  the  num- 
ber of  courses  in  each  being  the  same.  Find  the  number  of 
balls  in  the  lowest  course  of  the  triangular  pile. 

13.  If  from  a  complete  square  pile  of  ;^  courses  a  triangu- 
lar pile  of  the  same  number  of  courses  be  formed,  show  that 
the  remaining  balls  will  be  just  sufficient  to  form  another  tri- 
angular pile,  and  find  the  number  of  its  courses. 


LOGARITHMS.  1 93 


CHAPTER   XVI. 
LOGARITHMS. 

299.  The  Logarithm  of  a  number  is  the  exponent 
by  which  a  fixed  number,  called  the  base,  must  be 
affected  in  order  to  equal  the  given  number.  That 
is,  if  ^^  —  N,  X  v=>  the  logarithm  of  N  to  the  base  Uy 
which  is  written  jt  =  log,.  tV". 

Thus,  since  32  =    9,     2  =log3  9. 

Since  2^=16,     4  =  log^^  ^^• 

Since  10^=10,  lo^^  100,  lo' =  1000,  . . . , 

the  positive  numbers  i,  2,  3,  . , . ,  are  respectively  the  logarithms 
of  10,  100,  1000,  . . .,  to  the  base  10. 

To  the  base  10  the  logarithms  of  all  numbers  between  i  and 
10,  10  and  100,  100  and  1000,  ...,  are  incommensurable. 

Since         3"^^  =  ^      -2  =  Iog3f 

Since        10-1  =  0.1,  10-2  =  0.01,  10-' =  0.001,  .. ., 

the  negative  numbers  —1,  —2,  —3,  ...,  are  respectively  the 
logarithms  of  o.i,  o.oi,  o.ooi,  ...,  to  the  base  10. 

300.  Any  positive  number  except  I  may  evidently 
be  taken  as  the  base  of  logarithms. 

The  logarithms  of  all  positive  numbers  to  any 
given  base  constitute  a  System  of  Logarithms. 


194  ALGEBRA. 

In  any  system,  the  logarithms  of  most  numbers  are 
incommensurable. 

Before  discussing  the  two  systems  commonly  used, 
we  shall  prove  some  general  propositions  that  are 
true  for  any  system. 

301.  The  logarithm  of  i  is  0. 

For  tf°  =  I ;     .-.  log^  i  =  0. 

302.  The  logarithm  of  the  base  itself  is  i . 
For  a^  =:  a  ;     .-.  log^^;—  i. 

303.  The  logarithm  of  a  product  equals  the  sum  of 
the  logarithms  of  its  factors. 

Let  log«  M=  X,     log,,  N=y; 

then  J/=  a\  N^a\  §  299. 

Therefore  MN=a'---^-\ 

Hence        log«  {MN)  =x  +y  =  log, M  +  log^ JV. 

Similarly,  log.(i^iV^0=  log.^+  log.^+  log^Q; 
and  so  on,  for  any  number  of  factors. 

304.  The  logarithm  of  a  quotient  equals  the  loga- 
rithm of  the  dividend  fninus  that  of  the  divisor. 

Let  M=cf,    N=a''; 

then  M-^  N  =za''-''. 

Hence      loga  {M -^  N)  =  x  —  y  —  \og^  M  —  log« N. 


LOGARITHMS.  195 

305.  The  logarithm  of  a  positive  number  affected 
with  ariy  exponent  equals  the  logarithm  of  the  number 
multiplied  by  the  expofient. 

Let  M^a''; 

then,  whatever  be  the  value  of/, 

Hence         log,  {M^)  —px—p  log.,  M. 

306.  By  §  305,  the  logarithm  of  any  power  of  a 
number  equals  the  logarithm  of  the  number  multi- 
plied by  the  exponent  of  the  power ;  and  the  loga- 
rithm of  any  root  of  a  number  equals  the  logarithm 
of  the  number  divided  by  the  index  of  the  root. 

307.  From  the  principles  proved  above,  we  see 
that  by  the  use  of  logarithms  the  operations  of  multi- 
plication and  division  may  be  replaced  by  those  of 
addition  and  subtraction,  and  the  operations  of  in- 
volution and  evolution  by  those  of  multiplication  and 
division. 

Example.    Express  log«-^^!— -  in  terms  of  log<,  b,  log^  z^  \ogaX> 
Loga  ^^  =  log,  ^t  -  log,  (^2  ^f )  §  304. 

=  loga  di  -  (log,,  Z^  +  loga  X^)        §  303. 
=  |l0g„  6-2\QgaZ-^  loga  X.      §  305. 


196  ALGEBRA. 

308.  If  a  >  1,  and  ^*  =  iV; 

then  ii  JV >  I,  X  Is  positive; 
if  ^<  I,  X  is  negative; 

if  JV=  -x),  X  =  yo ; 

if  j\r=o,  x  =  ~yo. 

That  is,  if  the  base  is  greater  than  unity ^ 
(i.)    The  logarithm  is  positive  or  negative  according 
as  the  number  is  greater  or  less  tJian  nnity. 

(ii.)  The  logarithm  of  an  infinite  is  infinite ;  and 
the  logarithm  of  an  infinitesimal  is  a  nega- 
tive infinite,  or,  as  it  is  often  stated,  the 
logarithm  of  zero  is  negative  infinity, 

EXERCISE  36. 

1.  Find  log,  16;  log,  64;  loggSi;  log,  ^V ;  log^^; 
logs^V;   Io&ttV;   ^^^Z-o^hz)   logs  125. 

2.  If  10  is  the  base,  between  what  integral  numbers  does 
the  logarithm  of  any  number  between  i  and  10  He?  Of 
any  number  between  10  and  100?  Of  any  number  between 
100  and  1000?  Of  any  number  between  o.  i  and  i?  Of 
any  number  between  o.oi  and  0.1  ?  Of  any  number  between 
0.00 1  and  0.01  ? 

In  the  next  ten  examples  express  log«  y  in  terms  of  log«  by 
log,  c,  log^  X,  and  log,  z. 

3.  y=.z^b^,  5.  y^^^z^x'^. 


4.  J  —  y^2  ^  ^^^  6.  y  =  "s/z^x  .  ^zb~^ , 


LOGARITHMS.  1 97 

g.   x^  \  bz^  '.:  y^  :  xb^ , 
10.   x~^  '.  c^y^  \\  ^  '.  x^bi. 
z^     J        b_2^       V^ 


z^ 


^/x^ 


12.     2" 


Common  Logarithms. 

309.  Although  there  may  be  any  number  of  sys- 
tems of  logarithms,  there  are  in  general  use  only  two, 
the  Natural  and  the  Common.  The  Natural  system, 
called  also  the  Napierian^  from  Baron  Napier,  is  used 
for  analytical  purposes  only  ;  its  base  is  2.71828. 
The  Common  system  is  the  system  used  in  practical 
computations;  its  base  is  10.  It  was  introduced  in 
161 5  by  Briggs,  a  contemporary  of  Napier.    ' 

Both  Napierian  and  common  logarithms  are  writ- 
ten decimally.  Hereafter  when  no  base  is  written, 
the  base  10  is  understood. 

310.  From  the  equation  10*  =  A^  it  is  evident  that 
the  common  logarithms  of  most  numbers  consist  of 
an  integral  part  and  a  fractional  part. 


198  ALGEBRA. 

For  example,  2146  >  lo^  and  <  10*  ; 

.♦.  log  2146  =  3  +  a  decimal  fraction. 

Again,  0.04  >  iq-^  and  <  iq-^  ; 

.'.  log  0.04  =  —  2  +  a  decimal. 

The  integral  part  of  a  logarithm  is  called  the 
Characteristic,  and  the  decimal  part  the  Mantissa. 
For  convenience  in  the  use  of  common  logarithms, 
mantissas  are  always  made  positive.  Hence  the 
logarithm  of  any  number  less  than  unity  consists  of 
a  negative  characteristic  and  a  positive  mantissa. 

311.  The  characteristic  of  the  common  logarithm 
of  any  number  can  be  determined  by  one  of  the  two 
following  simple  rules : 

(i.)  If  the  number  is  greater  than  unity,  the  charac- 
teristic is  positive  a7id  numerically  one  less 
than  the  number  of  digits  iji  its  integral 
part. 

For  a  number  with  one  digit  in  its  integral  part 
lies  between  10^  and  lO^;  a  number  with  two  digits 
in  its  integral  part  lies  between  10^  and  10^;  and  so 
on.  Hence  if  N  denote  a  number  that  has  n  digits 
in  its  integral  part,  then  N  lies  between  10" ~'  and 
10";   that  is, 

jy  --    J  Q  (m  —  1)  +  a  fraction. 

.*.  logiV^=  {71  —  i)  +  a  mantissa. 

Thus,  log  2178.24  =  3  +  a  mantissa; 
log  3872416  =  6  +  a  mantissa. 


LOGARITHMS.  I99 

(ii.)  If  the  mimber  is  less  tha7t  tmityy  the  character- 
istic is  negative  and  numerically  one  greater 
than  the  number  of  ciphers  immediately  after 
the  decimal  point. 

For  a  decimal  with  no  cipher  immediately  after 
the  decimal  point  lies  between  10  "^  and  10^;  thus, 
0.327  lies  between  o.i  and  i  ;  a  decimal  with  one  ci- 
pher immediately  after  the  decimal  point  lies  between 
10 ~ 2  and  io~^;  thus,  0.0217  Hes  between  0.0 1  and 
0.1  ;  and  so  on.  Hence  if  Z^  denote  a  decimal  with  n 
ciphers  immediately  after  the  decimal  point,  then  D 
lies  between  10 -^''-^^^  and  io~";  that  is, 

^___    jQ  —  («+l)  +  a  fraction. 

.'.  log  /?  =  —  («  +  i)  +  a  mantissa. 

Thus  log  0.003217  =  —  3  +  a  mantissa  ; 
log  0.000081  =:  —  5  +  a  mantissa. 

The  converse  of  rules  (i.)  and  (ii.^  may  be  stated 
as  follows  : 

(i.)  If  the  characteristic  of  a  logarithm  is  +  n,  there 

are  n  4-  i  integral  places  in  the  corresponding 

ntimber. 
(ii.)  If  the  characteristic   is  —  n,    there  are  n  —  i 

ciphers  immediately  to  the  right  of  the  decimal 

point  ift  the  number. 

312.  I^g  {Ny,\o^'')=^\ogN  ±n.  §  303. 

Hence  if  «  is  a  whole  number,  log  N  and  log 
(A^  X  10-")  have  the  same  mantissa.     Therefore  if 


200  ALGEBRA. 

a  number  be  multiplied  or  divided  by  an  exact 
power  of  lo,  the  mantissa  of  its  logarithm  will  not 
be    changed. 

That  is,  the  common  logarithms  of  all  numbers  that 
have  the  same  sequence  of  significant  digits  have  the 
same  mantissa. 

Thus,  the  logarithms  of  21.78,  2178,  and  0.002178  have  the 
same  mantissa. 

313.  The  method  of  calculating  logarithms  will  be 
explained  in  §§  319,  322.  The  common  logarithms 
of  all  integers  from  i  to  200000  have  been  computed 
and  tabulated.  In  most  tables  they  are  given  to 
seven  places  of  decimals  ;  but  in  abridged  tables  they 
are  often  given  to  only  four  or  five  places.  Common 
logarithms  have  two  great  practical  advantages : 

(i.)  Characteristics  are  known  by  §  311,  so  that 
only  mantissas  arc  tabulated. 

(ii.)  Mantissas  are  determined  by  the  sequence  of 
digits  (§  312),  so  that  the  mantissas  of  inte- 
gers only  are  tabulated. 

When  the  characteristic  is  negative,  the  minus  sign 
is  written  over  the  characteristic,  to  indicate  that  the 
characteristic  alone  is  negative,  and  not  the  whole 
expression. 

Thus  3.845098,  the  logarithm  of  0.007,  is  equivalent  to  —  3 
+  0  845098,  and  must  be  distinguished  from  —3.845098,  in  whicli 
both  the  integral  and  decimal  part  are  negative. 


LOGARITHMS. 


201 


To  transform  a  negative  logarithm,  as  —3  26782,  so  that  the 
mantissa  shall  be  positive,  we  subtract  i  from  the  characteristic 
and  add  i  to  the  mantissa. 

Thus  —  3.26782  =  —  4  4-  (i  —  0.26782)  =  4.73218. 

To  divide  3.78542  by  5,  we  proceed  thus  : 

i  (3.78542) -H- 5 +  2.78542) 
=  1.55708. 

314.  For  logarithinic  tables  and  directions  in 
their  use,  the  student  is  referred  to  works  on  Trig- 
onometry. For  use  in  this  and  the  next  chapter 
we  give  below  the  common  logarithms  of  prime 
numbers  from   i   to   100. 


No. 

Logarithms. 

No. 
29 

Logarithms. 

No. 

Logarithms. 

2 

0.3010300 

1.4623980 

61 

,1.7853298 

3 

O.4771213 

31 

I.49I3617 

67 

1.8260748 

7 

0.8450980 

37 

1. 56820 1 7 

71 

I. 8512583 

II 

I. 041 3927 

41 

I. 6127839 

73 

I.S633229 

13 

I.I  139434 

43 

1.6334685 

79 

I.F97627I 

17 

1.2304489 

47 

1.6720979 

83 

1. 9190781 

19 

1.2787536 

53 

1.7242759 

89 

1.9493000 

23 

1. 3617278 

59 

1.7708520 

97 

1. 98677 1 7 

Log  5  =  log  (10 -^  2) 

=:  log  10  —  log  2 


0.30103  =  0.69897. 


In  like  manner  the  logarithms  of  all  integers  between  i  and 
100  can  be  obtained  from  those  given  in  the  table  above. 


202  ALGEBRA. 

The  utility  of  logarithms  in  facilitating  numerical 
computations  is  illustrated  by  the  following  example. 

.2  8 

Example.     Find   the   value   of  33  x  0.9^  ^  0.494,  given 
log  2.87686  =  0.458919. 

log  (3^  X  0.92  -  0.49^)  =  f  log  3  +  2  log  ^\  -  f  log  ^%\ 

=  flog3  +  2(log32-i)-|(log72-2) 
=  I  log  3  +  4  log  3  -  2  - 1  log  7  +  f 

=  ¥log3-|log7-i 

=  2.3265661  —  1.267647  —  0.5 

=  0.458919  =  log  2.8y686  ; 

.-.   3^  X  0.92  H-  0.49^  =  2.87686. 

EXERCISE  37. 

1.  Given  log  2659  =  3.424718;  find  log  26.59,  log 
0.2659,  log  265900,  log  0.0002659. 

2.  Given  log  2389  =  3.378216;  find  the  number  whose 
logarithm  is  1.378216,  0.378216,  2.378216,  5-378216, 
3.378216,    4.378216. 


Find  the 

common  logarithm  of 
8.    1.05. 

13- 

3.    84. 

Vo-oio5« 

4.   0.128. 

9.   0.0183. 
10.   0.02134. 

14. 
15. 

86^ 

5.    0.0125. 

V3S-^  27. 

6.    1.44, 

II.     ^^42^ 

16. 

4|. 

7.    1.06. 

12.    V374- 

17- 

25^. 

LOGARITHMS.  20$ 


l8.     .2I01T 


19.  0.015^  ^   V2  ^'    ^X2Tx^  ' 

20.  o-ooiS"^. 

3,.  0.63^.     24.  v5^i^.  ,6.  (---^-7)'. 

.JUL  '^^^  (0.002-^3)3 

22.    (14^15)^.  ^  ^^ 

27.  Find  the  seventh  root  of  0.00324,  given 

log  4409.2388  =  3-644363- 

28.  Find  the  eleventh  root  of  39-2-^,  given 

log  19.48445  =  1.2896883. 

29.  Find  the  product  of  3  7- 203,  3.7203,  0.0037203,  and 
372030;  given 

log  372.03  =  2.570578,  and  log  191.5631  =  2.282312. 

315.  Exponential  Equations.  An  exponential  equa- 
tion is  one  in  which  the  unknown  quantity  appears 
in  an  exponent.  Thus  2""  =  5,  b^''  ■\-  If  —  c,  and  x* 
=  10  are  exponential  equations.  Exponential  equa- 
tions are  solved  by  the  aid  of  logarithms. 
Example  i.  Solve  32-^43*=  54 ^r 2^+1, 
Taking  the  logarithms  of  both  members,  we  have 

2  ;i'  log  3  +  3  X  log  22  =  4  jf  log  5  +  (jir  +  I)  log  2  ; 
.'.     (2  log  3  +  6  log  2  -  4log  5  -  log  2)  ;r  =  log  2, 
^log2 


or  x  = 


2log3  +  slog2-4logs 

0.301030 


0.336488 


0.894-}-. 


204  ALGEBRA. 

Example  2.    Find  the  logarithm  of  32/^/4  to  the  base  2y'2. 

Let  X  =  log  32y^4  to  base  2^2,  or  22  ; 

then  (2^)''  =  32^4  =  2*^  X  25 

■^  2 

Hence   "^  ;ir  log  2  =  $  log  2  +     log  2 ; 
2  5 

27       3       18 
.*.     x=  —  -^  ^  =  —  =  3.6. 
525-^ 

Example  3.    Solve  32-^  —  14  x  3""  +  45  =  0. 
The  equation  may  be  written  in  the  form 
(3"  -  9)  (3"  -  5)  -  0, 
which  is  equivalent  to  the  two  equations 
r  =  9  and  3-  =  5. 
From  y  —  g,  X  =  2]  and  from  y  =  5, 

^^  logs  ^0698970^  ^5 
log  3      0.477121 

EXERCISE  38. 

Solve  the  following  literal  equations  : 

2.    ^2x  ^8^  _  ^5^  4.    ^  =  <r«^ 

Solve  the  following  numerical  equations,  using  the  table 
in  §  314: 

5.  5^:^800.  8.    23-52  — i.=  45-3-  +  i. 

6.  5^-8  =  82--^*.  9.    2-6^-2=  52^  7^-^. 

7.  12- =  3528.  10.    4'"+  56  =  15  X  4". 


LOGARITHMS.  20$ 

Find  the  logarithm  of 

11.  1 6  to  base  V2  ,  and  1728  to  base  2  VS' 

12.  125  to  base  5  Vs  ,  and  0.25  to  base  4. 

13.  5^^  to  base  2  ^2  ,  and  0.0625  to  base  2 . 

14.  Find  log,  128;  logg^}^;  log^^^V- 

Solve  the  system  of  equations 

15.  x>-^f,  16.    a'''b''y  =  m\ 
x^  =y\ 


a^-  f'y  =  m^\ 


Logarithmic  and  Exponential  Series. 

316.    The  Derivative  of  log^^.      Let 

j>  =  nz,  (i) 

71  being  an  arbitrary  constant,  and  y  and  js  functions 
of  X. 

Then  log^^  =  log,,  ft  +  log,,  z ; 

.-.  A(log.>')-Z>,(log.0).  (2) 

Dividing  the  derivatives  of  the  members  of  (i)  by 
(i),  we  obtain 

^  =  ^^  (3) 

Dividing  (2)  by  (3)  we  obtain, 

D,  (I0&7)  :  ^  =  ^.  (log.  ^)  :  ^-       (4) 
y  z 


206  ALGEBRA. 

It  is  evident  that  the  equal  ratios  in  (4)  are  constant 
for  any  particular  value  of  z.  Let  m  denote  their  con- 
stant value  when  z  =  z  ; 

then  D,  (log,^)  =  m  ^,  (5) 

when y  =.  n^.  But  as  n  is  an  arbitrary  constant,  71  ^ 
denotes  any  number;  hence  (5)  holds  true  for  all 
values  of  y,  m  being  a  constant. 

The  constant  m  is  called  the  Modulus  of  the  system 
of  logarithms  whose  base  is  a. 

Hence,  the  derivative  of  the  logarithm  of  a  variable 
is  equal  to  the  modulus  of  the  system  into  the  derivative 
of  the  variable  divided  by  the  variable. 

EXERCISE  39. 
Find  the  derivative  of 

1.  J  =  log,  (i  +  ^).  4-  /W  =  (log«^)^ 

2.  7=  log.  {x^  +  oc^)'  5-  /W  =  ioga-^^- 

3.  ^  =  log,  (^2  +  ^  +  b).  6.  fix)  =  X  log,  X, 


'J.  y  =  log„  Vi—x^=i  log«  (i  —  x^). 

S.  y  =  log,  (x^  +  x)2.  10.  y  =  log,  (x^  —  cx^)t 

,  X  Vi  +  -^ 

9.  y  =  \og,--==z.  II.  j;  =  log,— =.r. 

yi  +  x^  yi  —  X 

317.    To  deduce  the  Logarithmic  Series. 

To  do  this,  we  develop  log,  (i  ■\-  x)  by  Maclaurin's 
formula, 


/'(.v)  = 

i+x' 

f>\x)  = 

m 

(i+xf' 

/'"(*)  = 

2  m 

(i+A)" 

f\x)  = 

|3«» 

(i+^r' 

LOGARITHMS.  207 

/W  -/(O)  4-/'(0)  Y  +  /"  (0)  ^'  +/'"(0)  ^'  +/'M0)  ^  + . . . 

Here     /(^)  -  log,  (i  +  x\     .-.       /(O)  -  0 ; 

..    ///(O)  =_;;,. 

.    /iv(0)=_i3^, 


Substituting  these  values  in  the  formula,  we  have 
lofc(i+^)  =  «(x-^%^-^V^-...),      (A) 
which  is  the  general  logarithmic  series, 

318.  Napierian  System.  The  system  of  logarithms 
whose  modulus  is  unity  is  called  the  Napierian  or 
Natural  system.  The  symbol  for  the  Napierian  base 
is  e. 

If  in  (A)  of  §  317  we  put  m  =  i  and  a  =  e,  we 
have 

^2  ^,3  „4  „5 

log,  (I  +  a:)  =  ^  --  +  ---  +  --...  ,  (B) 

2345 

which  is  the  Napierian  logarithmic  series. 


208  ALGEBRA. 

By  Example  i  of  §  286  the  series  in  (B)  is  con> 
vergent  only  for  values  of  x  between  —  i  and  +  i ; 
hence  formula  (B)  cannot  be  used  to  compute  the 
Napierian  logarithm  of  any  number  greater  than  2. 

319.    To  obtain  a  formula  for  computing  a  table  of 
ISlapierian  logarithms. 

Putting  —  ;!;  for  ;ir  in  (B)  of  §  3 18,  we  have 


(0 


log    (l+^)-l0g,(l-.T)  =  2(^  +  -  +  ^  +  -+...).(2) 


Let  ^  =  TT-r-;;  (3) 


log.  (I  -:.)=- 

—  X  — 

x" 
2 

x^ 
3 

^4 
4 

x' 
'  5 

Subtracting  (i) 

from 

(B), 

we 

have 

then 


22+1' 
I  +  ^      z  -^  1 


1  —  X 


.'.  log,  (i  +  x)  -  log,  (i  -  ^)  =  log,  (2:  +  i)  -  log,  z.     (4) 

Substituting  in  (2)  the  values  in  (3)  and  (4),  we 
obtain 

l0g,(^+l)=l0g.^+2(^  +  ^^j^+|y^^,  +  ...).(C) 

Since,  in  (3),^<  i  for  ^>0,  the  series  in  (C)  is  con- 
vergent for  all  positive  values  of  ^;  hence  log,  {2  +  1) 
can  be  readily  computed  when  log,  z  is  known. 


LOGARITHMS.  20g 

Example.  Compute  to  six  places  of  decimals  logv2,  log^3, 
log,  4,   log,  5,   log,  lo. 

Putting  ^  =  I  in  (C),  we  obtain,  since  log^  i  =  0, 

Summing  six  terms  of  this  series,  we  find 

log,  2  =  0.693147. 
Putting  z  =  2'in  (C),  we  have 

=  1. 0986 1 2. 

Log,  4  =  2  log,  2  =  1.386294. 
Putting  2-  =  4  in  (C),  we  obtain 

=  1.6094379. 

Log,  10  =  log,  5  +  log,  2  =  2  302585. 

In  this  way  the  Napierian  logarithms  of  all  positive  numbers 
can  be  computed.  The  larger  the  number  the  more  rapidly 
convergent  is  the  series. 

320.    Value  of  m.     Dividing  (A)  by  (B),  we  have 
log,,  (i  +x) .. 

in  which  i  +  x  lies  between  0  and  2. 
Let  N  be  any  number,  and  let 

V  =  locr.  JV,  or  A/'=  a^ : 


2IO 

ALGEBRA. 

then 

\og,JV=y  .  log,  a  =  log^iV^.  log,  a. 

Hence 

log^JV          I                                       ,  . 

LetiV^  = 

I  +  ;r;  then  from  (i)  and  (2),  we  have 

««  =  ,-7:^-                          (3) 

That  is,  t/2e  modulus  of  any  system  of  logarithms  is 
equal  to  the  reciprocal  of  the  Napierian  logarithm  of  its 
base, 

321.  From  (2)  and  (3)  of  §  320  we  have 

log,,  N=^  7)1  log,  N. 

That  is,  the  logarithm  of  a  number  in  a7ty  system 
is  equal  to  the  Napieria7i  logaritlnn  of  the  same  number 
multiplied  by  the  modulus  of  that  sysiem. 

322.  Value  of  M.  If  in  (3)  of  §  320,  M  denote  the 
value  oim  when  a  —  10,  we  obtain 

Jf  =  ;— =  IT  =  0-4342Q4. 

log,  10        2.302585  ^^^  ^^ 

That  is,  the  modulus  of  the  Common  System  to 
six  places  of  decimals  is  0.434294. 

Hence  to  obtain  common  logarithms  from  Na- 
pierian, multiply  the  Napierian  by  0.434294;  to  ob- 
tain Napierian  from  common,  multiply  the  common 
by  2.302585,  or  log,  10. 


LOGARITHMS.  2 1 1 

Multiplying  both  members  of  (C)  by  M,  we  obtain 

which  is  a  formula  for  computing  common  logarithms. 
Multiplying  both  numbers  of  (C)  by  ;//,  we  obtain 
a  general  formula  for  computing  logarithms  to  any 
base  a, 

323.  An  Exponential  Function  is  one  in  which  the 
variable  enters  the  exponent,  as  ^^X^  o!"^'"* 

324.  To  find  the  derivative  of  a^ 

Let  y  =^  a' )  then  log^^  =  zlog^a,  (i) 

Hence  -^  =  log,  a  -  Z>^z, 

y 

or  D^y  =  A  («~~)  =  a''  log,«  •  A  z- 

That  is,  t/ie  derivative  of  an  exponeiitial  function 
with  a  constant  base  is  equal  to  the  fu7iction  itself  into 
the  Napierian  logarithm  of  the  base  into  the  derivative 
of  the  exponent. 

325.  To  develop  a*,  or  deduce  the  Expojiential  Series, 

Here     f{x)  =  (f,  .-.    /(O)  =  i ; 

fix)  =  or  \og,a,  .-.  /HO)  =  \og,a; 

f\x)  =  cf  (log.«)^  .-.  f\Q)  =  Qog^ay ; 

f"(x)  =  a^  {\og,a)\  ,'.f"{0)  =  (log,«)« ; 


212  ALGEBRA. 

Substituting  these  values    in  Maclaurin's  formula, 
we  have 

a'=i  +  (Xog^a)  X  +  {Xog^af  ^  +  {\og,aff-  +  ...  ,    (i) 

Li  11 

which  is  the  exponential  series, 

326.  Value  of  e^ .        Putting  ^  =  ^  in  (i)  of  §  325, 
we  have 

^•2  ^8  ^4 

327.  Value  of  e.      Putting  x=\    in  (i)  of  §  326, 
we  obtain 

^=  1  +  I  +  P  +  i-  +  ^  + •••  =  2.718281. 
That  is,  the  Napierian  base  —  2.7 1828 1  +.* 


EXERCISE   40. 

1.  Find  to  five  places  log^ 6,  log^  7,  log^ 8,  log^  9,  log^  1 1. 

2.  Find  to  five  places  the  moduli  of  the  systems  whose 
bases  are  2,  3,  4,  5,  8,  9,  12. 

3.  By  §  321,  prove  that  the  logarithms  of  the  same  num- 
ber in  different  systems  are  proportional  to  the  moduH  of 
those  systems. 

*  In  the  "  Proceedings  of  the  Royal  Society  of  London,"  Vol. 
XXVII.,  Prof.  J.  C.  Adams  has  given  the  values  of  e,  M,  log,  2,  log,  3, 
log^  5,  to  more  than  260  places  of  decimals. 


LOGARITHMS.  21 3 

4.  By  the  formula  of  §  322  compute  log  2,  log  65,  log  131, 

log  3;  log  82,  log  244. 

5.  Obtain  the  formula, 

log.  (.  +  I)  =  log..  +  .  m  (^^  +  ^^^l^^y  +•••)• 

6.  Obtain  the  following  formulas : 

log.(^+  '^"^''^^""^"^2  +  ^8 W 

log.^  -  log.(0  -  0  =  ^  +  2-72  +  ^8  +  •••  (2) 

log,  (.  +  I)  -  log,  (._  1)  =  2  (i  +  ^'^  +  ^  +  ...).  (3) 

These  formulas  are  convergent  for  0  >  i,  and  may  be 
used  in  computing  logarithms. 

To  obtain  (i),  in  (B)  substitute  (i  —z)  ioxx;  to  obtain  (2), 
substitute  (  —  i  -H  2)  for  ^. 

7.  Obtain  the  formulas  corresponding  to  (i),  (2),  and  (3) 
of  Example  6,  for  common  logarithms  :  for  any  system. 

4/-  lOy- 

8.  Show  that  log  -Y^    —_  =  i  log  5  -  f  log  2  -  §  log  3. 

V 18  .  V2 

9.  Show  that  log  y  729  y  9~~^X  27    ^  =  log3. 

10.  Find  the  logarithms  ofi/^s-^  —=,  \/^~^  >  to  base  a. 

11.  Find  the  number  of  digits  in  3^^  X  2^ 


214  ALGEBRA. 

AiNioo 

12.  Show  that  I — 1       is  greater  than  I oo. 

13.  Find  how  many  ciphers  there  are  between  the  deci- 
mal point  and  the  first  significant  digit  in  (^)^'^°^ 

14.  Calculate  to  six  decimal  places  the  value  of 


m 

X    I25\2 
X32    J     ' 

;iven  log  9076.226 

=  3-95  79< 

^53- 

Solve 

15.    2^+-^  =6-^, 

16.   3^— > 

=  4-^ 

s^'^sx 

2^  +  \ 

22.-1 

=  33.-^ 

17.    If  log  {x^ y^)  =  a,  and   log(^  -J- 7)  =  d,  find   log  x 
and  log  J. 


o     o,         ,         hmit 
18.   Show  that 

m  = 


unit    /         xK*" 

f         x\*^  X      7n(m—\)lx\'^ 

V        tn)  tn  p         \7nj 

[2  [3 


limit  /        :r\'' 

I  +-)  rz  i+;r  +  — +  l:^  +  •••  =  ^• 


limit  (        x\^  x^     x^ 

19.    If  «  is  positive,  the  positive  real  value  of  a''  is  a  con- 
tinuous function  of  x. 

For  ^'^  has  one  positive  real  value  for  each  value  of  x,  and 
A(«^)  =  a^  (a^  —  i)  =  0  when  A^  =  0. 


COMPOUND   INTEREST.  21 5 


CHAPTER   XVII. 

COMPOUND  INTEREST  AND  ANNUITIES. 

328.  To  find  the  interest  I  and  ainoimt  M  of  a  given 
sunt  P  /;/  n  years  at  r  per  centj  compoimd  interest. 

(i.)    When  interest  is  payable  anmially. 

Let  R^=  the  amount  of  $  i  in  i  year;  then  R  =  i 
+  r,  and  the  amount  of  P  at  the  end  of  the  first  year 
is  PR;  and  since  this  is  the  principal  for  the  second 
year,  the  amount  at  the  end  of  the  second  year  is 
PR  X  R,  or  P R^.  For  hke  reason  the  amount  at 
the  end  of  the  third  year  is  P R^,  and  so  on;  hence 
the  amount  in  n  years  is  P  R" ;  that  is 

M=  PR",  or  P{i  +  ry,  (i) 

Hence  I=P(R"-\).  (2) 

(ii.)    When  the  interest  is  payable  q  times  a  year. 

If  the  interest  is  payable  semi-annually,  then  the 
interest  of  $1  for  i  a  year  is  \r ;  hence 

the  amount  of  P  in  \  a  year  is  P(i  +  2  r); 
the  amount  of  P  in  one  year  is  P  (i  +  J  r)^ ; 
the  amount  of  P  in  ;/  years  is  /*  (i  +  |  r)^". 


2l6  ALGEBRA. 

That  is,  M=F{Y-\-\rf\  (3) 

Similarly,  if  the  interest  is  payable  quarterly, 

M=P{Y^\ry\  (4) 

Hence,  if  the  interest  is  payable  q  times  a  year, 


G+^r       (5) 


M^  P 

9 

In  this  case  the  interest  is  said  to  be  "converted 

into  principal  "  q  times  a  year. 

Example.  Find  the  time  in  which  a  sum  of  njoney  will 
double  itself  at  10  per  cent  compound  interest,  interest  payable 
semi-annually. 

Here  i  ■\-\r  ~  1.05. 

Let  P  =  $i;    then  M  =  $2. 

Substituting  these  values  in  (3),  we  obtain 

2=  (i.05)2«; 

.*.     log  2.—  in  '  log  1.05  ; 

loof  2 


2  (log  5  +  log  3  +  log  7  -  2) 

o  30^03 

■~  2  (0.69897  +  0.477 1 21 3  -I-  0.845098  —  2) 

=  7-103. 
Therefore  the  time  is  7.103  years. 

329.    When  the  time  contains  a  fraction  of  a  year^ 
it  is  usual  to  allow  simple  interest  for  the  fraction  of 

the  year.     Thus  the  amount  of  P  in    11  -\ years  is 

m 


COMPOUND   INTEREST.  21/ 

PR^  +  PR'^  -,  or  PR"  (i  +  -V 

When  interest  is  payable  oftener  than  once  a  year 
there  is  a  difference  between  the  nominal  annual  rate 
and  the  true  anmtal  rate.  Thus,  if  interest  is  payable 
semi-annually  at  the  nominal  annual  rate  r,  the  amount 
of  ;^l  in  one  year  is  (i  +  \  r)^,  ox  \  -\-  r-\-\  r^,  so  that 
the  true  annual  rate  is  r  +  J  r^. 

Thus,  if  the  nominal  annual  rate  is  4  per  cent,  and  interest  is 
payable  semi-annually,  the  true  annual  rate  is  4.04  per  cent. 

330.  Present  Value  and  Discount.  Let  P  denote 
the  present  value  of  the  sum  J/ due  in  ;/  years,  at 
the  rate  r ;  then  evidently,  in  ;/  years,  at  the  rate  r, 
P  will  amount  to  M ;  hence 

M^PR'\  or  P=MR-''. 

Let  D  be  the  discount ;  then 

£>  =  M-P=M{i  -R-"). 

EXERCISE  41. 

1.  Write  out  the  logarithmic  equations  for  finding  each  of 
the  four  quantities  M,  R,  P,  n. 

2.  In  what  time,  at  5  per  cent  compound  interest,  will 
$100  amount  to  $1000? 

3.  Find  the  time  in  which  a  sum  will  double  itself  at 
4  per  cent  compound  interest. 


21 8  ALGEBRA. 

4.  Find  in  how  many  years  $1000  will  become  $2500  at 
10  per  cent  compound  interest. 

5.  Find  the  present  value  of  $10,000  due  8  years  hence 
at  5  per  cent  compound  interest;  given  log  67683.94  = 
4.8304856. 

6.  Find  the  amount  of  $1  at  5  per  cent  compound 
interest  in  a  century;    given  log  13 15  =1=3.11893. 

7.  Show  that  money  will  increase  more  than  seventeen- 
thousand-fold  in  a  century  at  10  per  cent  compound  inter- 
est, interest  payable  semi-annually;  given  log  17213.13  = 
4.23786. 

8.  Find  what  sum  of  money  at  6  per  cent  compound 
interest  will  amount  to  $1000  in  12  years;  given  log  49697 
=  4.6963292,  log  106  =  2.0253059. 

9.  Find  the  amount  of  a  cent  in  200  years  at  6  per  cent 
compound  interest;  given  log  115. 128  =  2.06118. 

10.  The  present  value  of  $672  due  in  a  certain  time  is 
$126  ;  if  compound  interest  at  4^  per  cent  be  allowed,  find 
the  time. 

ANNUITIES. 

331.  An  Annuity  is  a  fixed  sum  of  money  that  is 
payable  once  a  year,  or  at  more  frequent  regular 
intervals,  under  certain  stated  conditions.  An  Aft- 
miity  Certain  is  one  payable  for  a  fixed  number  of 
years.     A  Life  Annuity  is  one  payable  during  the 


ANNUITIES.  219 

lifetime  of  a  person.  A  Perpetual  Annuity,  or  Per- 
petuity, is  one  that  is  to  continue  forever,  as,  for  in- 
stance, the  rent  of  a  freehold  estate.  A  Deferred 
Ajifiuity  is  one  that  does  not  begin  until  after  a 
certain  number  of  years. 

332.  To  find  the  amount  of  an  annuity  left  unpaid 
for  a  given  number  of  year's,  allowing  compound  in- 
terest. 

Let  A  be  the  annuity,  n  the  number  of  years,  R 
the  amount  of  one  dollar  in  one  year,  M  the  required 
amount.  Then  evidently  the  sum  due  at  the  end  of 
the 

1st  year  =  A, 

2d  year  =  A  R  •\-  A. 

3d  year  ^  A  R""  +  A  R -^  A. 

ni\i  yedix  =  A  R'-''  +  A  R"-""  +  ...  +  A  R  +  A 

_  A  (R'  -  i) 
'~      R-  I      " 

That  is,       M=^(R"-i).  (i) 

Example  i.  Find  the  amount  of  an  annuity  of  $100  in 
20  years,  allowing  compound  interest  at  4.^  per  cent;  given 
log  1.045  =  0.0 191 163,  log  24.117  =  1.382326. 

r  ^  ^  0.045 

By  logarithms         1 .04520  =  2.41 1 7 ; 

0.045        *^-J  -5' 


22(3  ALGEBRA. 

Example  2.  Find  what  sum  must  be  set  aside  annually  that 
it  may  amount  to  $50,000  in  10  years  at  6  per  cent  compound 
interest  ;  given  log  17.9085  =  1.253059. 

Solving  (i)  for  A  we  obtain 

Mr        $  50,000  X  0.06 

By  logarithms  1.0610=1.79085; 

$  3000 

333.  To  find  tJie  present  value  of  an  annuity  to  con- 
tinue for  a  given  number  of  years  y  allowing  compound 
interest. 

Let  P  denote  the  present  value ;  then  the  amount 
o{  P  in  n  years  will  equal  the  amount  of  the  annuity 
in  the  same  time ;   that  is, 

PR"  =  ^{R"-^)i  (i) 

'.■.P=-^{x-R-).  (2) 

334.  Perpetuity.  If  the  annuity  be  perpetual,  then 
n  =  -Xi,  R'""  ==  0,  and  (2)  of  §  333  becomes 

r 

335.  Deferred  Annuity.  If  the  annuity  commences 
after  /  years,  and  continues  7t  years  thereafter,  then 
the    present  value   will   evidently  be    the    difference 


ANNUITIES.  '  221 

between  the  present  value  of  an  annuity  to  continue 
n  ^r  p  years  and  one  to  continue/  years;  that  is, 

P=-^{R-^-Rr''-%  (i) 

„      A      R'-z. 


336.  If  the  annuity  be  perpetual  after/  years,  then 
R~"-^=  0,  and  (i)  of  §  335  becomes 

F=-  R-K 
r 

337.  Solving  (i)  of  §  333   for  ^,  we  obtain 
Fr  7?" 


A  = 


R" 


which  gives  the  value  of  the  annuity  in  terms  of  the 
present  value,  the  time,  and  the  rate  per  cent. 

338.  A  Freehold  Estate  is  an  estate  which  yields  a 
perpetual  annuity,  called  rent;  hence  the  value  of 
the  estate  is  the  present  value  of  a  perpetuity  equal 
to  the  rent. 

Example  i.     Find  the  present  value  of  an  annual  pension 

of  $200  for  10  years  at  5  per  cent  interest ;  given  log6. 13917 

=  0.78S107. 

A  $  200 

P=-{i-R--)  =  - (I  ^1.05-10). 

r^  /       0.05   ^  ^       '' 

By  logarithms  1.05- ^^  =:  0.613917  ; 

.-.    F  =  $4000  X  0.386083  =  $  1544.33. 


222  ALGEBRA. 

Example  2.  The  rent  of  a  freehold  estate  is  $350  a  year. 
Find  the  value  of  the  estate,  the  rate  of  interest  being  5  per  cent. 

A      $350 
F  =  —  =  ■ — ^  =  $  7000. 
r       0.05       ^ ' 

Example  3.  Find  the  present  value  of  an  annuity  of  %  1400 
to  begin  in  8  years  and  to  continue  12  years,  at  8  per  cent  in- 
terest;  given  log  25.1818  rr  14010868,  log  466.1  =  2.668478. 

^      A       R"  -  \       $1400       1.0812-1 

^  =  7  "^  1F+}  =  "^.^  ^  ~7:o8"^o-  =  $5700.09. 


Example  4.     Find  what  annuity  $  5000  will  give  for  6  years 
when  money  is  worth  6  per  cent;  given  log  14.185  =  1. 15 18344. 

^       Pr/e«  1.066 

A  =  j^,^  _  ^  =  %  5000  X  0.06  X  ^^Q^6_i  -  $  1016.84. 


EXERCISE   42. 

1.  If  A  leaves  B  $1000  a  year  to  accumulate  for  3  years 
at  4  per  cent  compound  interest,  find  what  amount  B  should 
receive;  given  log  112.4864  =  2.05 11 062. 

2.  Find  the  present  value  of  the  legacy  in  Example  i ; 
given  log  888.9955  —  2.9488998. 

3.  Find  the  present  value,  at  5  per  cent,  of  an  estate  of 
$1000  a  year,  (i)  to  be  entered  on  immediately,  (2)  after 
3  years;  given  log  17376.75  —  4.2374621. 

4.  A  freehold  estate  worth  $120  a  year  is  sold  for  $4000 ; 
find  the  rate  of  interest. 


ANNUITIES.  223 

5.  A  man  borrows  $5000  at  4  per  cent  compound  inter- 
est; if  the  principal  and  interest  are  to  be  repaid  by  10 
equal  annual  instalments,  find  the  amount  of  each  instal- 
ment; given  log  675560  =  5.829666. 

6.  A  man  has  a  capital  of  $20,000,  for  which  he  receives 
interest  at  5  per  cent;  if  he  spends  $1800  every  year,  show 
that  he  will  be  ruined  before  the  end  of  the  1 7th  year. 

7.  When  the  rate  of  interest  is  4  i)er  cent,  find  what  sum 
must  be  paid  now  to  receive  a  freehold  estate  of  $400  a  year 
10  years  hence  ;  given  log  6.75560  =  0.829666. 

8.  The  rent  of  a  freehold  estate  of  $882  per  annum, 
deferred  for  two  years,  is  to  be  sold ;  find  its  present  value 
at  5  per  cent  compound  interest. 

9.  The  rent  of  a  freehold  estate,  deferred  for  6  years,  is 
bought  for  $20,000 ;  find  what  rent  the  purchaser  should 
receive,  reckoning  compound  interest  at  5  per  cent;  given 
log  1.340096  =  0.1271358. 

10.  Find  the  present  worth  of  a  perpetual  annuity  of  $  10 
payable  at  the  end  of  the  first  year,  $  20  at  the  end  of 
the  second,  $  30  at  the  end  of  the  third,  and  so  on,  increas- 
ing $  10  each  year,  interest  being  taken  at  5  per  cent  per 
annum. 


24  ALGEBRA. 


CHAPTER  XVIII. 

PERMUTATIONS   AND    COMBINATIONS. 

339.  Fundamental  Principle.  If  one  thing  can  be 
done  in  m  ways,  and  {after  it  has  been  done  in  ajty 
one  of  these  ways)  a  second  thing  can  be  done  in  n 
ways ;    then   the  two   things  can   be   done   in  m  X  n 

ways. 

After  the  first  thing  has  been  done  in  any  one  way, 
the  second  thing  can  be  done  in  n  different  ways ; 
hence  there  are  ;/  ways  of  doing  the  two  things  for 
each  of  the  m  ways  of  doing  the  first ;  therefore  in  all 
there  are  m  n  ways  of  doing  the  two  things. 

Ihis  principle  is  readily  extended  to  the  case  in 
which  there  are  three  or  more  things,  each  of  which 
can  be  done  in  a  given  number  of  ways. 

Example  i.  If  there  are  ii  steamers  plying  between  New 
York  and  Havana,  in  how  many  ways  ran  a  man  go  from  New 
York  to  Havana  and  return  by  a  different  steamer  ? 

He  can  make  the  first  passage  in  ii  ways,  with  each  of  which 
he  has  the  choice  of  lo  ways  of  returning  ;  hence  he  can  make 
the  two  journeys  in  ii   x   lo,  or  no,  ways.  ^ 

Example  2.  In  how  many  ways  can  3  prizes  be  given 
to  a  class  of  10  boys,  without  giving  more  than  one  to  the 
same  boy  ? 


PERMUTATIONS    AND   COMBINATIONS.  225 

The  first  prize  can  be  given  in  lo  w.iys  ;  with  each  of  which 
the  second  prize  can  be  given  in  9  ways  ;  hence  the  first  two 
prizes  can  be  given  in  10  x  9  ways.  With  each  of  these  ways 
tlie  third  prize  can  be  given  in  8  ways;  hence  the  three  prizes 
can  be  given  in  10  x  9  x  8,  or  720,  ways. 

340.  Each  of  the  different  groups  of  r  things  which 
can  be  made  of ;/  things  is  called  a  Combination. 

The  Permutations  of  any  number  of  things  are  the 
different  orders  in  which  they  can  be  arranged,  taking 
a  certain  number  at  a  time. 

Thus  of  the  four  letters  a,  b,  c,  d,  taken  two  at  a  time,  there 
are  six  combinations  ;  namely, 

ab^    ac^    ad^    be,    bd,    cd. 

Each  of  these  groups  can  be  arranged  in  two  different  orders; 
hence  of  the  four  letters  a.b,  c,  d,  taken  two  at  a  time  there 
are  tv.elve  permutations  ;  namely, 

ab,     ac,     ad,     be,     bd,     ed, 
ba,     e  a,     da,     eb,     db,     de. 

Of  a  group  of  three  letters,  7is  abe,  when  taken  all  at  a  time, 
there  are  six  permutations  ;  namely, 

a  be,     aeb,     be  a,     bae.     cab,     eba. 

The  symbol  "P^  will  be  used  to  denote  the  number 
of  permutations  of  71  things  taken  r  at  a  time. 

Thus,  ^P^,  ^P^,  ®/*4,  denote  respectively  the  number  of  per- 
mutations of  9  things  taken  2,  3,  4,  at  a  time. 

Similarly  "C,.  will  be  used  to  denote  the  number  of 
combinations  of  ;/  things  taken  r  at  a  time. 


226  ALGEBRA. 

341.  To  find  the  number  of  permutations  of  n  dis- 
similar things  taken  v  at  a  time. 

The  number  required  is  the  same  as  the  number  of 
ways  of  fining  r  places  with  ;/  things. 

Now,  the  first  place  can  be  filled  by  any  one  of  the 
n  things,  and  after  this  has  been  filled  in  any  one  of 
these  n  ways,  the  second  place  can  evidently  be  filled 
in  {n  --  i)  ways;  hence  with  n  things  two  places  can 
be  filled  in  n  (it  —  i )  ways ;  that  is, 

-F^  =  n(7i-i).  (i) 

After  the  first  two  places  have  been  filled  in  any 
one  of  these  n{n  —  i)  ways,  the  third  place  can  be 
filled  in  (ji  —  2)  ways ;  hence  three  places  can  be 
filled  in  ;/  (n  —  i)  (ji  —  2)  ways ;   that  is, 

"Pg  —  n{n—  i)  {11  —  2).  (2) 

For  like  reason  we  have 

«/>,  z=n{n-  i)  {11  -2){n~i)\  (3) 

and  so  on. 

From  (i),  (2),  (3),  ...,  we  see  that  in  "P^  there 
are  r  factors,  of  which  the  rth  is  ;^  —  r  +  i ;   hence 

^P^  ^n  {n—  i)  {n  —  2)  ...  {?i  —  r -\-  i).  (A) 

342.  Value  of  "P„.     U  r  —  n,  (A)  of  §  341  becomes 

"^.  =  \n.  (B) 

That  is,  the  number  of  permutations  of  n  things 
t.:ken  all  at  a  time  is  \n. 


PERMUTATIONS  AND   COiMBINATIONS.  22/ 

343.  Circular  Permutations.  When  n  different  let- 
ters are  arranged  in  a  circle,  any  one  of  their  per- 
mutations can  without  change  be  revolved  so  that 
any  letter,  as  a,  shall  have  a  given  position.  Hence 
we  may  regard  a  as  having  the  same  position  in  all 
the  permutations.  Now  -the  number  of  the  possible 
arrangements  of  the  remaining  ;/  —  i  letters  in  the 
other  positions  is  \n—  I . 

Hence,  the  number  of  the  Circular  Permutations  of 
n  things  is  \n—\. 

EXERCISE  43. 

1.  A  cabinet-maker  has  12  patterns  of  chairs  and  7  pat- 
terns of  tables.  In  how  many  ways  can  he  make  a  chair 
and  a  table?  Ans.   84. 

2.  There  are  9  candidates  for  a  classical,  8  for  a  mathe- 
matical, and  5  for  a  natural-science  scholarship.  In  how 
many  ways  can  the  scholarships  be  awarded? 

3.  In  how  many  ways  can  2  prizes  be  awarded  to  a  class 
of  10  boys,  if  both  prizes  may  be  given  to  the  same  boy? 

4.  Find  the  number  of  the  permutations  of  the  letters  in 
the  word  numbers.  How  many  of  these  begin  with  fi  and 
end  with  s  ? 

5.  If  no  digit  occur  more  than  once  in  the  same  number, 
how  many  different  numbers  can  be  represented  by  the  9 
digits,  taken  2  at  a  time  ?  3  at  a  time  ?  4  at  a  time  ? 


228  ALGEBRA. 

6.  How  many  changes  can  be  rung  with  5  bells  out  of  8? 
How  many  with  the  whole  peal  ? 

The  first  number  =  ^P^  =  6720. 

7.  How  many  changes  can  be  rung  with  6  bells,  the  same 
bell  always  being  last? 

8.  In'  how  many  ways  may  a  host  and  6  guests  be  seated 
at  a  table  in  a  row?  In  how  many  ways  if  the  host  must 
have  Mr.  Jones  on  his  right  and  Mr.  Smith  on  his  left?  In 
how  many  ways  if  the  host  must  sit  between  Mr.  Smith  and 
Mr.  Jones? 

9.  In  how  many  ways  may  15  books  be  arranged  on  a 
shelf,  the  places  of  2  being  fixed? 

10.  Given  "F^  =  12  .  '^7^2 ;  find  n. 

11.  Given  n  :  ""F^  : :  i  :  20;  find  n, 

12.  In  how  many  different  orders  may  a  party  of  6  be 
seated  at  a  round  table? 

13.  In  how  many  different  orders  may  10  persons  form 
a  ring? 

14.  In  how  many  different  orders  may  a  host  and  8 
guests  sit  at  a  round  table,  provided  the  host  has  Mr.  A  at 
his  right  and  Mr.  B  at  his  left? 

15.  Given  "F^  :  "  +  ^^3  :  :  5  :   12,  to  find  n. 

16.  Given  'F^  :  ^"F^  :  :   13  :  2,  to  find  n. 


PERMUTATIONS   AND    COMBINATIONS.  229 

344.  To  find  the  number  of  combinations  of  n  dis- 
similar things  taken  r  at  a  time. 

By  §  342  there  are  \r  permutations  of  any  com- 
bination of  r  things;  hence  we  have 

=  n  (n  —  1)  («  —  2)  ...  («  —  r  +  i). 
Hence    -C  =  "<"-')("- g""  ^^ -"+'>  .  (C) 

345.  Corollary  i.  Multiplying  the  numerator 
and  denominator  of  the  fraction  in  (C)  by  \n  —  r,  we 
obtain 

_n{n-i)  {n-2)  ...  {n-r+i)  \n  -  r 
[r  \n-r 


or        «C=i— pl= — .  (D) 

Formula  (C)  should  be  used  when  a  numerical 
result  is  required.  In  applying  this  formula,  it  is 
useful  to  note  that  the  suffix  r  in  the  symbol  "C^ 
denotes  the  number  of  the  factors  in  both  the  nu- 
merator and  denominator  of  the  formula.  Formula 
(D)  gives  the  simplest  algebraic  expression  for  "C^. 

346.  Corollary  2.  Substituting  n  —  r  for  r  in 
(D)  we  obtain 

"C,._r=,  '-      ,      .  (l) 

\n  —  r  \r  ^ 

From  (D)  and  (i),  "C  =  "C.-..  (E) 


230  ALGEBRA. 

The  relation  in  (E)  follows  also  from  the  con- 
sideration that  for  each  group  of  r  things  that  is 
selected,  there  is  left  a  corresponding  group  of  ;/  —  r 
things.  This  relation  often  enables  us  to  abridge 
arithmetical  work. 

Thus,  ^^^13  =  15^2  =  ^^^^==105. 

EXERCISE   44. 

1.  How  many  combinations  can  be  made  of  9  things  taken 

4  at  a  time  ?  taken  6  at  a  time  ?  taken  7  at  a  time  ? 

The  last  number  =  ^Q  =  ^Cg  =  36. 

2.  How  many  combinations  can  be  made  of  11  things 
taken  4  at  a  time?   taken  7  at  a  time? 

3.  Out  of  10  persons  4  are  to  be  chosen  by  lot.  In  how 
many  ways  can  this  be  done  ?  In  all  the  ways,  how  often 
would  any  one  person  be  chosen? 

4.  From  14  books  in  how  many  ways  can  a  selection  of 

5  be  made,  (i)  when  one  specified  book  is  always  included, 
(2)  when  one  specified  book  is  always  excluded? 

5.  On  how  many  days  might  a  person  having  15  friends 
invite  a  different  party  of  10?  of  12? 

6.  Given  ^"C^  =  15,  to  find  71. 

7.  Given  "  +  ^C;  =  9  X  ''C^,  to  find  71, 

8.  In  a  certain  district  there  are  4  representatives  to  be 
elected,  and  there  are  7  candidates.  How  many  different 
tickets  can  be  made  up  ? 


PERMUTATIONS  AND   COMBINATIONS.  23  I 

9.  Of  8  chemical  elements  that  will  unite  with  one  another, 
how  many  ternary  compounds  can  be  formed  ?  How  many 
binary  ? 

10.  On  a  table  are  6  Latin,  7  Greek,  and  8  German  books. 
In  how  many  different  ways  may  2  books  from  different 
languages  be  chosen  ?     In  how  many  ways  may  3  ? 

The  first  number  =6x7  +  6x8  +  7x8  =  146. 

11.  In  how  many  ways  can  10  gentlemen  and  10  ladies 
arrange  themselves  in  couples? 

12.  How  many  different  arrangements  of  6  letters  can  be 
made  of  the  26  letters  of  the  alphabet,  2  of  the  5  vowels 
being  in  every  arrangement? 

13.  How  many  different  straight  lines  can  be  drawn 
through  any  15  points,  no  3  of  which  lie  in  the  same  straight 
line? 

14.  In  a  town  council  there  are  25  councillors  and  10 
aldermen  ;  how  many  committees  can  be  formed,  each 
consisting  of  5  councillors  and  3  aldermen? 

15.  Find  the  sum  of  the  products  of  the  numbers  3,  —  2, 
4»  ~  5»  i»  (0  taken  2  at  a  time,  (2)  taken  3  at  a  time, 
(3)  taken  4  at  a  time. 

16.  Find  the  sum  of  the  products  of  the  numbers  i,  3,  5, 
2,  (i)  taken  2  at  a  time,  (2)  taken  3  at  a  time. 

17.  Find  the  number  of  combinations  of  55  things  taken 
45  at  a  time. 

18.  If  ^"Ca  :  "Cj  =  44  :  3  ;  find  n. 


232  ALGEBRA. 

19.  If  "C,2  =  "Q ;  find  "C,. ;  find  ^''C^, 

20.  Ill  a  library  there  are  20  Latin  and  6  Greek  books; 
in  how  many  ways  can  a  group  of  5  consisting  of  3  Latin 
and  2  Greek  books  be  placed  on  a  shelf  ? 

21.  From  3  capitals,  5  other  consonants,  and  4  other 
vowels,  how  many  permutations  can  be  made,  each  begin- 
ning with  a  capital  and  containing  in  addition  3  consonants 
and  2  vowels? 

22.  Ifi«C  =  ''C+2;  find'-Q. 

23.  From  7  Englishmen  and  4  Americans  a  committee  of 
6  is  to  be  formed ;  in  how  many  ways  can  this  be  done 
when  the  committee  contains,  (i)  exactly  2  Americans,  (2)  at 
least  2  Americans? 

24.  Of  7  consonants  and  4  vowels,  how  many  permutations 
can  be  made,  each  containing  3  consonants  and  2  vowels  ? 

25.  How  many  different  arrangements  can  be  made  of 
the  letters  in  the  word  courage,  so  that  the  consonants 
may  occupy  even  places? 

347.  //"N  de7iote  the  mimber  of  permutations  of  x\ 
things  taken  all  at  a  time,  of  which  r  things  are  alike, 
s  others  alike,  and  t  others  alike  ;  then 


N= 


\L\L\L 


Suppose  that  in  any  one  of  the  A^  permutations  we 
replaced  the  r  like  things  by  r dissimilar  things;  then, 


PERMUTATIONS   AND    COMBINATIONS.  233 

from  this  single  permutation,  without  changing  in  it 
the  position  of  any  one  of  the  other  n  —  r  things,  we 
could  form  V  new  permutations.  Hence  from  the 
^original  permutations  we  could  obtain  NW  permu- 
tations, in  each  of  which  s  things  would  be  alike  and 
t  others  alike. 

Similarly,  if  the  s  like  things  were  replaced  by  s 
dissimilar  things,  the  number  of  permutations  would 
be  iVIr  !.r,  each  having  /  things  alike.  Finally,  if  the 
t  like  things  were  replaced  by  t  dissimilar  things  we 
should  obtain  N\r\s  \t_  permutations,  in  which  all  the 
things  would  be  dissimilar. 

But  the  number  of  permutations  of  «  dissimilar 
things  taken  all  at  a  time  is  \n. 

Hence  iV^[r  [f;  [£  =  |«. 

\n 
Therefore  N 


348.  To  find  the  number  of  ivays  in  which  m  4-  n 
things  caji  be  divided  into  two  groups  containing  re- 
spectively  m  and  n  things. 

The  number  required  is  evidently  the  same  as  the 
number  of  combinations  of  m  +  n  things  taken  m  at 
a  time  ;  for  every  time  a  group  of  i7t  things  is  selected 
a  group  of ;/  things  is  left. 

Hence  the  required  number  =  ■ ■  §  345. 


234  ALGEBRA. 

349.  By  §  348  the  number  of  ways  in  which  m  +  n 
+  /  things  can  be  divided  into  two  groups  containing 
respectively  m  and  n  -\-  p  things  is 

\m  +  ^i  -\-  p 
\ni  \n  +/ 

Again,  the  number  of  ways  in  which  each  group  of 
n  -\-  p  things  can  be  divided  into  two  groups  con- 
taining respectively  n  and  /  things  is 

V^  +  P 
\n\p_ 

Hence  the  number  of  ways  in  which  in  \  n  ■\-  p 
things  can  be  divided  into  three  groups  containing 
respectively  ;;/,  ;/,   and  p  things  is 

\m  ■\- n  ■\- p       \n  +  p  \m -\- n  ■\- p 

\m^\Ti  -\-  p  (^  \P  '  \m\71\p 

This  reasoning  can  be  extended  to  any  number  of 
groups. 

*350.    The  sum  of  all  the  combinations  that  can  be 
made  of  n  things,  taken  i ,  2,  . . .  ,  w  at  a  time^  is  2"  —  i . 
By  the  binomial  theorem  we  have 

/         X  n{f^ — i)    ^     n(n — 1)(«— 2)    ,  ,  . 

(i+;v)«=:i  +  ^^^+-h ^-x^^^ f^ ^a:H  -    (i) 

\^  [3 

In  (i)  the  coefficients  of  x,  x^,  x^,  ...,  x"'  are  evi- 
dently the  values  of  "Q,  "C2,  "Q,  ...,  "C„;  hence  (i) 
may  be  written 

(i+xy=i  +  "C,x  +  "C2x''-{-"Qx^+'^'  +"C„x\         (2) 


PERMUTATIONS   AND   COMBINATIONS.  235 

Putting  X  =  I,  and  transposing  i,  we  obtain 
2"  _  I  =.  "C,  +  "C,  +  "Q  +  ...  +  "C«, 

which  proves  the  proposition. 

*351.    "Cr  is  greatest  wJien  r=^n^rr  =  Kn  ±  i), 

according  as  n  is  even  or  odd. 

I  n 
Evidently  "C>  or  \ — ^= — »  is  greatest  when  \r\n  —  r 
^  \r\n  —  r  ■-  ' 

is  least. 

Since  \a  -\-  i  [^  — 'i  is  obtained  from  \a  \a  by  mul- 
tiplying hy  a  -\-  I  and  dividing  by  a,  it  follows  that 

Irt  1^  <  1^  +  I    \a~  \    <  |«  +  2   1^  —  2  <  ... 

Hence  when  n  is  even,  \r  \n  —  r  is  least,  and  there- 
fore "Cr  is  greatest,  when  r=  71  —  r,  or  r  =  |  ;/. 

Again  |^  \fi -\-  i  =  |/^  +  i  |^ 

and  |/^+  I  [^  <  1^4-  2  |/^  -  i  <  |/^  +  3  |/^-2  <  ... 

Hence  when  n  is  odd,  Ir  |«  —  r  is  least,  and  there- 
fore "Cr  is  greatest,  when  r  =  n  —  r  ±  i,orr=|(« 
±  I). 

EXERCISE   45. 

1.  How  many  different  arrangements  can  be  made  of  the 

letters  of  the  word  commencement  ? 

Of  the  12  letters,  2  are  ^'s,  3  are  m^s,  3  are  <?'s,  and  2  are  ;/'s  ; 

|I2 

,-.  JV=  - — ,'-—    ,    =  3326400. 

[2[3[3[2 

2.  Find  the  number  of  permutations  of  the  letters  of  the 
words,  mammalia,  caravansera,  Mississippi. 


236  ALGEBRA. 

3.  In  how  many  ways  can  17  balls  be  arranged,  if  7  of 
them  are  black,  6  red,  and  4  white? 

4.  When  repetitions  are  allowed,  "F^  =  «%  and  "/^„  =  n". 
When  repetitions  are  allowed  after  the  first  place  has  been 

filled  in  any  one  of  fi  ways,  the  second  place  can  be  filled  in  ;/ 
ways  ;  hence    "P^  =  n%  etc. 

5.  In  how  many  ways  can  4  prizes  be  awarded  to  10  boys, 
each  boy  being  eligible  for  all  the  prizes? 

6.  At  an  election  three  districts  are  to  be  canvassed  by  10, 
15,  and  20  men,  respectively.  If  45  men  volunteer,  in  how 
many  ways  can  they  be  allotted  to  the  different  districts  ? 

7.  In  how  many  ways  can  52  cards  be  divided  equally 
among  4  players? 

8.  In  how  many  ways  can  m  n  things  be  divided  equally 
among  n  persons? 

9.  How  many  signals  can  be  made  by  hoisting  4  flags  of 
different  colors  one  above  the  other,  when  any  number  of 
them  may  be  hoisted  at  once  ?     How  many  with  5  flags  ? 

10.  There  are  25  points  in  space,  no  4  of  which  lie  in 
the  same  plane.  Find  how  many  planes  there  are,  each 
containing  3  points. 

11.  For  what  value  of  r  is  i°C  greatest?  "C?  ^^C? 
20c;?  31c:,? 

12.  For  what  value  of  r  is  [r  |i8  —  r  least?  \r  [21  —  r? 

13.  What  term  has  the  greatest  coefficient  in  the  develop- 
ment of  {x  +  7)"  ?  {x  +  >')"  ?  i^  +  yf''  ? 


PROBABILITY.  23/ 


CHAPTER  XIX. 

PROBABILITY. 

352.    If  an  event  may  happen  in  a  ways  and  fail  in 

b  ways,  and  each  of  these  ways  is  equally  likely,  the 

a 
Probability,  or  the  Chance,  of  its  happening  is  — — :, 

b  ^"^^ 

and  the  probjbi'tty  of  its  failing  is -. 

Hence  to  find  the  probability  of  an  event,  divide 
the  Jiimibcr  of  cases  that  favor  it  by  the  whole  Jiumber 
of  cases  for  and  against  it. 

For  example,  if  in  a  lottery  there  are  5  prizes  and  22  blanks, 
the  probability  that  a  person  holding  i  ticket  will  win  a  prize  is 
^,  and  the  probability  of  his  not  winning  is  |^. 

Example  i.  From  a  bag  containing  8  white,  7  black,  and 
5  red  balls,  one  ball  is  drawn.  Find  the  chance,  (i)  that  it  is 
white,  (2)  that  it  is  black  or  red. 

In  all  there  are  20  ways  of  drawing  a  ball ;  of  these  20  ways 
8  are  favorable  to  drawing  a  white  ball,  and  12  to  drawing  a 
black  or  a  red  ball;  hence  the  chance  of  the  ball  being  white 
is  -^Q,  or  |,  and  that  of  its  being  black  or  red  is  f . 

Example  2.  From  a  bag  containing  7  white  and  4  red  balls, 
3  balls  are  drawn  at  random.  Find  the  chance  of  these  being 
all  white. 

The  whole  number  of  ways  in  which  3  balls  can  be  drawn  is 
"Cg;  and  the  number  of  ways  of  drawing  3  white  balls  is  'Cg; 
therefore,  of  drawing  3  white  balls 

,     ,  'Q       7-6.5       7 

the  chance  =  r^  -  — —  =  —  • 

"Cg      II.  10. 9      33 


238  ALGEBRA. 

353.    Unit  of  Probability.      The  sum  of  the  proba- 

.  .  .  a  b       .  . 

bihties and is  unity;   hence  if  the  prob- 

a  ^  b  a  +  b  ^  ^ 

abihty  that  an  event  will  happen  is/,  the  probability 
that  it  will  fail  is  i  — /. 

If  b  is  zero,  the  event  is  certain  to  happen,  and  its 
probability  is  unity ;  hence  certainty  is  the  tuiit  of 
pi'obability. 

Instead  of  saying  that  the  probability  of  an  event 

a 

is 7,  we  sometimes  say  that  the  odds  are  a  /^  b  in 

a  -\-  b 

favor  of  the  event,  or  h  to  a.  agaijist  it. 

Example  i.  Find  the  chance  of  throwing  at  least  one  ace 
in  a  single  throw  with  three  dice. 

Here  it  is  simpler  to  first  find  the  chance  of  not  throwing  an 
ace.  Each  die  can  be  thrown  in  five  ways  so  as  not  to  give  an 
ace  ;  hence  the  three  can  be  thrown  in  5^,  or  125,  ways  that  will 
exclude  aces  (§  339).  The  total  number  of  ways  of  throwing  3 
dice  is  6^,  or  216.  Hence  the  chance  of  not  throwing  one  or 
more  aces  is  125  -^  216 ;  so  that  the  chance  of  throwing  at  least 
one  ace  is  i  -  \\%  or  ^^Ve  (§  353)- 

Here  the  odds  against  the  event  are  125  to  91. 

Example  2.  A  has  3  shares  in  a  lottery  in  which  there  are 
4  prizes  and  7  blanks;  B  has  i  share  in  a  lottery  in  which  there 
is  I  prize  and  10  blanks  ;  show  that  A's  chance  of  success  is  to 
B's  as  26  is  to  3. 

A  can  get  all  blanks  in  "^Cg,  or  35,  ways ;  he  can  draw  3  tick- 
ets in  "Cg,  or  165,  ways  ;  hence  A's  chance  of  failure  —  /g^  =  jg- 
Therefore  A's  chance  of  success  =  i  —  /j  =  f |. 

B's  chance  of  success  is  evidently  ^^\ 

.'.  A's  chance  :  B's  chance  =  ff  :  i\  =  26  :  3. 


PROBABILITY.  239 

Or  to  find  A's  chance  we  may  reason  thus  :  A  may  draw  3  prizes 
in  ^Cg ,  or  4,  ways  ;  he  may  draw  2  prizes  and  i  blank  in  *C^  x  7, 
or  42,  ways;  he  may  draw  i  prize  and  2  blanks  in  4  x  'Q,  or 
84,  ways  ;  hence  A  can  succeed  in  4  +  42  +  84,  or  130,  ways. 
Therefore  A's  chance  of  success  =  y^  =  |f. 

EXERCISE   46. 

1.  From  the  vessel  on  which  Mr.  A  took  passage  one 
person  has  been  lost  overboard.  There  were  60  passengers 
and  30  in  the  crew.  Find,  (i)  the  chance  that  Mr.  A  is 
safe,  (2)  the  chance  that  all  the  passengers  are  safe,  (3)  the 
probability  that  a  passenger  is  lost. 

2.  There  are  15  persons  sitting  around  a  table  ;  find  the 
probability  that  any  2  given  persons  sit  together. 

Wherever  one  of  the  2  persons  sits,  the  other  may  occupy 
any  one  of  14  places,  of  which  2  will  put  the  2  persons  to- 
gether. 

3.  According  to  the  Carlisle  Table  of  Mortality,  it  appears 
that  out  of  6335  persons  living  at  the  age  of  14  years,  only 
6047  reach  the  age  of  2 1  years.  Find  the  probability  that  a 
child  aged  14  years  will  reach  the  age  of  21  years.  Find  the 
chance  that  he  will  not  reach  it. 

4.  From  a  bag  containing  4  red  and  6  black  balls,  2  balls 
are  drawn;  find  the  chance,  (i)  that  both  are  red,  (2)  that 
both  are  black,  (3)  that  one  is  red  and  the  other  black. 

5.  From  a  bag  containing  4  white,  5  black,  and  6  red 
balls,  3  balls  are  drawn  ;  find  the  probability  that  (i)  all  are 
white,  (2)  all  black,  (3)  all  red,  (4)  2  black  and  i  red,  (5)  i 
white  and  2  black.  Ans.    ^f^,  ^,  ^\,  H»  /t- 


240  ALGEBRA. 

6.  When  two  coins  are  thrown,  find  the  chance  that  the 
result  will  be,  (i)  both  heads,  (2)  both  tails,  (3)  head  and 
tail. 

7.  When  two  dice  are  thrown,  what  is  the  probability  of 
throwing,  (i)  a  5  and  6,  (2)  two  6's? 

8.  ;^rom  a  committee  of  7  Republicans  and  6  Democrats, 
a  sub-committee  of  3  is  chosen  by  lot.  What  is  the  proba- 
bility that  it  will  be  composed  of  2  Republicans  and  i 
Democrat? 

,  9.  From  a  committee  of  8  Democrats,  7  Republicans,  and 
3  Independents,  a  sub-committee  of  4  is  chosen  by  lot. 
Find  the  chance  that  it  will  consist,  (i)  of  2  Democrats  and 
2  Republicans,  (2)  of  i  Democrat,  2  Republicans,  and  i 
Independent,  (3)  of  4  Democrats. 

10.  In  a  single  throw  with  two  dice,  show  that  the  chance 
of  throwing  5  is  l- ;  of  throwing  6  is  /^. 

IT.  One  of  two  events  must  happen,  and  the  chance  of 
the  first  is  two  thirds  that  of  the  second ;  find  the  odds  in 
favor  of  the  second. 

12.  In  a  bag  are  4  white  and  6  black  balls;  find  the 
chance  that,  out  of  5  drawn,  2,  and  2  only,  shall  be  white. 

13.  In  Example  12  show  that  the  chance  of  2  at  least 
being  white  is  %\. 

14.  Out  of  TOO  mutineers,  a  general  orders  two  men,  cho- 
sen by  lot,  to  be  shot ;  the  real  leaders  of  the  mutiny  being 
TO,  find  the  chance  that,  (i)  one  of  the  leaders  will  be  taken, 
(2)  two  of  them. 


PROBABILITY.  24 1 

15.  A  has  3  shares  in  a  lottery  containing  3  prizes  and  9 
blanks ;  B  has  2  shares  in  a  lottery  containing  2  prizes  and 
6  blanks  ;  compare  their  chances  of  success. 

16.  There  are  4  half-dollars  and  3  quarter-dollars  placed 
at  random  in  a  line ;  prove  that  the  chance  of  the  extreme 
coins  being  both  quarter-dollars  is  f.  In  the  case  of  in  half- 
dollars  and  11  quarter-dollars,  show  that  the  chance  is 

n{n-\) 
{m  -\-  n){m  -\-  n  —  i) 

17.  There  are  three  works,  one  consisting  of  3  volumes, 
another  of  4,  and  tlie  third  of  i  volume.  They  are  placed 
on  a  shelf  at  random ;  prove  that  the  odds  against  the  vol- 
umes of  the  same  works  being  all  together  are  137  to  3. 

18.  A  man  wants  a  particular  span  of  horses  from  a  stud 
of  8.  His  groom  brings  him  5  horses  taken  at  random. 
What  is  the  chance  that  both  horses  of  the  span  are  among 
them  ? 

19.  Of  the  three  events  A,  B,  C,  one  must,  and  only  one 
can,  occur ;  A  can  occur  in  a  ways,  B  in  b  ways,  and  C  in  ^ 
ways,  all  the  ways  being  equally  likely ;  find  the  chance  of 
each  event. 

Compound  Events. 

354,  Thus  far  we  have  considered  only  single 
events.  The  concurrence  of  two  or  more  events  is 
sometimes  called  a  Compound  event. 

Events  are  said  to  be  Dependent  or  Independent^ 
according  as  the  happening  (or  failing)  of  one  event 
does  or  does  not  affect  the  occurrence  of  the  other. 


242  ALGEBRA. 

355.  The  probability  that  two  independeitt  events 
will  both  happen  is  equal  to  the  product  of  their  sepa- 
rate probabilities. 

Suppose  that  the  first  event  may  happen  in  a  ways 
and  fail  in  b  ways,  all  these  cases  being  equally  likely ; 
and  suppose  that  the  second  event  may  happen  in  a^ 
ways  and  fail  in  b^  ways,  all  these  cases  being  equally 
likely.  Each  of  the  a  -{■  b  cases  may  be  associated 
with  each  of  the  a^  +  b'  cases  to  form  {a  +  b)  {ci  +  U) 
compound  cases,  all  equally  likely  to  occur. 

In  a  a'  of  these  compound  cases  both  events  hap- 
pen, in  by  of  them  both  fail,  in  ab^  of  them  the  first 
happens  and  the  second  fails,  and  in  a^  b  of  them  the 
first  fails  and  the  second  happens.     Hence 

7 ;^v^i 77T  ===  the  chance  that  both  events  happen  ; 

{a  +  b)  {a}  +  ^0 

bb' 

=  the  chance  that  both  events  fail ; 


(a  +  b)  (af  +  b') 

a  b^  f  the  chance  that  the  first  happens  and 


=1 


{a  +  b)  (a'  +  b')        \      the  second  fails ; 

a'  b  { the  chance  that  the  first  fails  and  the 


+  ^')        1 


(a  +  b)  (a'  +  b^)        \      second  happens. 

356.  The  probability  that  any  number  of  independent 
events  will  all  happen  is  equal  to  the  product  of  their 
separate  probabilities. 

Let  ply  p2,  and /a  be  the  respective  probabilities  of 
three  independent  events.  The  probability  of  the 
concurrence  of  the  first  and  second  events  is  pi  p2\ 


PROBABILITY.  243 

the  probability  of  the  concurrence  of  the  first  two 
events  and  the  third  is  {pipi)  pz,  ox p^p,p^\  and  so 
on  for  any  number  of  events. 

357.  By  §  356,  if  /  is  the  chance  that  an  event  will 
happen  in  one  trial,  the  chance  of  its  happening  each 
time  in  71  trials  is/". 

358.  The  chance  that  all  three  of  the  events  in 
§356  will  fail    is  (I  -A)  (I  -/,)  (I  -/a). 

Hence  the  chance  that  some  one  at  least  of  them 
will  happen    is    i  —  (i  — /i)  (i  —  A)  (»  —  Pz)- 

The  chance  that  the  first  two  will  happen  and  the 
third  fail  is  AA(i  -/a). 

Example  i.  Find  the  chance  of  throwing  an  ace  in  the  first 
only  of  2  successive  throws  with  a  single  die. 

The  chance  of  throwing  an  ace  in  the  first  throw  =  \. 
The  chance  of  not  throwing  an  ace  in  the  second  throw  =  |. 
Hence  the  chance  of  the  compound  event  =  J  x  ^  =  ^. 

Example  2.  From  a  bag  containing  6  white  and  9  black 
balls,  2  drawings  are  made,  each  of  3  balls,  the  balls  first 
drawn  being  replaced  before  the  second  trial  ;  find  the  chance 
that  the  first  drawing  will  give  3  white,  and  the  second  3  black 
balls. 

The  number  of  ways  of  drawing  3  balls  =  ^^q  . 
"  "  "  "  3white  =  6<^;; 

3  black  =  9(73. 
Hence,  of  drawing  3  white  balls  at  first  trial 

the  chance  =  ..^  =  '  ^  '  ^     =  ~- . 

i^Cg       15.  14.  13       91' 


65* 

4         I2_ 

48 
5915 

244  ALGEBRA. 

and,  of  drawing  3  black  balls  at  second  trial 

the  chance  =  -p-^  =  —^ — ^-^— 
i-^Cg       15.  14.  13 

Hence  the  chance  of  the  compound  event: 


Example  3.  If  the  odds  are  1 1  to  9  against  a  person  A,  who 
is  now  38  years  old,  living  till  he  is  68,  and  4  to  3  against  a 
person  B,  now  43,  living  till  he  is  73  ;  find  the  chance  that  one 
at  least  of  these  persons  will  be  alive  30  years  hence. 

The  chance  that  A  will  die  within  30  years  =  il  ;  the  chance 
that  B  will  die  within  30  years  =  -f ;  hence  the  chance  that  both 
will  die  =  ^l  X  ^  =  II ;  therefore  the  chance  that  both  will  ?wl 
die,  that  is,  that  one  at  least  will  be  alive,  =  i  —  -i|  ==  ||. 

Example  4.  In  how  many  trials  will  the  probability  of 
throwing  an  ace  with  a  single  die  amount  to  |? 

Let  X  =  the  number  of  trials.  By  §  357,  the  chance  of  failing 
to  throw  an  ace  each  time  in  x  trials  is  (l)"-  Hence  the  chance 
of  throwing  an  ace  once  at  least  in  x  trials  is    1  —  (|)^ ; 

.-.     i-(l)^  =  ior(i-)-  =  ^; 

•      X  =         ^"g3         ^  0-4771213  ^  g^^^ 
log  6  — log  5      0.0791813 

Hence  in  6  trials  the  chance  of  success  is  a  little  less  than  |, 
and  in  7  trials  it  is  greater  than  |. 

359.  Dependent  Events.  A  slight  modification  of 
the  meaning  of  ^  and  <^'  in  §  355  enables  us  to  estimate 
the  chance  of  the  concurrence  of  two  dependent  events. 
Thus,  if  after  the  first  event  has  happened,  a'  denote 
the  number  of  ways  in  which  the  second  can  follow, 
and  U  the  number  of  ways  in  which  it  will  not  fol- 
low;   then  the  number  of  cases  in  which  the  two 


PROBABILITY.  245 

events    will  both  happen  is  aa\  and  the  chance  of 

,    .  .  ^  ^' 

their   concurrence   is rr^—. 77-. 

{a  +  /;)  {a'  +  b') 

Hence  if/  is  the  chance  of  the  first  of  two  depen- 
dent events,  and  /'  the  chance  that  the  second  will 
follow,  the  chance  of  .their  concurrence  is  //'. 

Example.  From  a  bag  containing  6  white  and  9  black 
balls,  two  drawings  are  made,  each  of  3  balls,  the  balls  first 
drawn  not  being  replaced  before  the  second  trial  ;  find  the 
chance  that  the  first  drawing  will  be  3  white  and  the  second 
3  black  balls. 

At  the  first  trial,  3  balls  may  be  drawn  in  ^^Cg  ways  ;  and 
3  white  balls  may  be  drawn  in  ^C^  ways  ;  hence  the  chance 

!^  =     ^-5-4     ^  4_ 
'C3      15.14.13      91- 

After   3  white  balls  have  been   drawn  the  bag  contains  3 

white  and  9  black  balls  ;  therefore,  at  the  second  trial,  3  balls 

may  be  drawn  in  ^^Q  ways  ;  and  3  black  balls  may  be  drawn  in 

®6"g  ways ;  hence,  of  drawing  3  black  balls  at  the  second  trial, 

,       ,  »C.         9-8.7         21 

the  chance  =  ttt^  = =  — . 

i^Cg      12  .  II  .  10      ss 

Hence  the  chance  of  the  compound  event  =  ^|-  x  |^  =  -jSos- 
The  student  should  compare  this  result  with  that  of  Example 
2  in  §  358. 

EXERCISE  47. 

1.  Show  that  the  chance  of  throwing  an  ace  in  each  of 
two  successive  throws  with  a  single  die  is  ^jr. 

2.  Show  that  the  chance  of  throwing  an  ace  with  a  single 
die  in  two  trials  is  \^. 


of  3  white  balls  at  the  first  trial  =  j. 


246  ALGEBRA. 

3.  A  traveller  has  5  railroad  connections  to  make  in  order 
to  reach  his  destination  on  time.  The  chances  are  3  to  i 
in  favor  of  each  connection.  What  is  the  probability  of  his 
making  them  all? 

4.  Mr.  A  takes  passage  on  a  ship  for  London.  The 
probability  that  the  ship  will  encounter  a  gale  is  -^.  The 
probability  that  she  will  spring  a  leak  in  a  gale  is  ^.  In  case 
of  a  leak,  the  probability  that  the  engines  will  be  able  to 
pump  her  out  is  f.  If  they  fail,  the  probability  that  the 
compartments  will  keep  her  afloat  is  £.  If  she  sinks,  it  is 
an  even  chance  that  any  one  passenger  will  be  saved  by  the 
boats.  What  is  the  probability  that  Mr.  A  will  be  lost  at 
sea  on  the  voyage? 

5.  In  how  many  trials  will  the  chance  of  throwing  an  ace 
with  a  single  die  amount  to  ^  ? 

Ans.    In  4  trials  the  chance  is  a  little  greater  than  ^. 

6.  The  odds  against  A's  solving  a  certain  problem  are  i 
to  2,  and  the  odds  in  favor  of  B's  solving  the  same  problem 
are  3  to  4 ;  find  the  chance  that  the  problem  will  be  solved 
if  they  both  try. 

7.  The  chance  that  a  man  will  die  within  ten  years  is  ^, 
that  his  wife  will  die  is  ^,  and  that  his  son  will  die  is  ^ ;  find 
the  chance  that  at  the  end  of  ten  years,  (i)  all  will  be  living, 
(2)  all  will  be  dead,  (3)  one  at  least  will  be  living,  (4) 
husband  living,  but  wife  and  son  dead,  (5)  wife  living,  but 
husband  and  son  dead,  (6)  husband  and  wife  living,  but 
son  dead. 


PROBABILITY.  247 

8.  A  bag  contains  2  white  balls  and  4  black  ones.  Five 
persons,  A,  B,  C,  D,  E,  in  alphabetical  order  each  draw 
one  ball  and  keep  it.  The  first  one  who  draws  a  white  ball 
is  to  receive  a  prize.  Show  that  their  respective  chances  of 
winning  are  as  5  :  4  :  3  :  2  :  i. 

A's  chance  of  winning  the  prize  is  easily  obtained. 

That  B  may  win,  A  must  fail.  Hence  to  find  B's  chance, 
we  find,  (i)  the  chance  that  A  fails,  (2)  the  chance  that  if  A 
fails  B  will  win.     We  then- take  the  product  of  these  chances. 

That  C  may  win,  (i)  A  must  fail,  (2)  B  must  fail,  (3)  C 
must  draw  a  white  ball.  Hence  C's  chance  of  winning  is  the 
product  of  the  chances  of  these  three  events;  and  so  on. 

9.  A  and  B  have  one  throw  each  of  a  coin.  If  A  throws 
head,  he  is  to  receive  a  prize ;  if  A  fails  and  B.  throws  head, 
he  is  to  receive  the  prize.  If  A  and  B  both  fail,  C  receives 
the  prize.     Find  the  chance  of  each  man  winning  the  prize. 

10.  From  a  bag  containing  5  white  and  8  black  balls  two 
drawings  are  made,  each  of  3  balls,  the  balls  not  being 
replaced  before  the  second  drawing ;  find  the  chance  that 
the  first  drawing  will  give  3  white  and  the  second  3  black 
balls. 

11.  In  three  throws  with  a  pair  of  dice,  find  the  chance 
of  throwing  doublets  at  least  once. 

12.  Find  the  chance  of  throwing  6  with  a  single  die  at 
least  once  in  5  trials. 

13.  The  odds  against  a  certain  event  are  6  to  3,  and  the 
odds  in  favor  of  another  event  independent  of  the  former  are 
7  to  5  ;  find  the  chance  that  one  at  least  of  the  events  will 
happen. 


248  ALGEBRA. 

14.  A  bag  contains  17  counters  marked  with  the  numbers 
T  to  1 7.  A  counter  is  drawn  and  replaced  ;  a  second  draw- 
ing is  then  made  ;  find  the  chance  that  the  first  number 
drawn  is  even  and  the  second  odd. 

*  360.  If  a7i  evefit  can  happen  m  two  or  more  different 
ways,  which  are  mutually  exclusive,  the  cha^tce  that  it 
will  happen  is  the  sum  of  the  chances  of  its  happening 
in  these  different  zvays. 

When  these  different  ways  are  all  equally  probable, 
the  proposition  is  merely  a  repetition  of  the  definition 
of  probability.  When  they  are  not  equally  probable, 
the  proposition  is  often  regarded  as  self-evident  from 
that  definition.  It  may,  however,  be  proved  as 
follows : 

Let  -r  and  ~  be  respectively  the  chances  of  the 

happening  of  an  event  in  two  ways  that  are  mutually 
exclusive.  Then  out  of  b^  hi  cases  there  are  a^  b-i  cases 
in  which  the  event  may  happen  the  first  way,  and 
a^  bi  cases  in  which  the  event  may  happen  the  second 
way;  and  these  ways  are  mutually  exclusive.  There- 
fore out  of  bi  b^  cases,  ai  b^  +  a^  b^  cases  are  favorable 
to  the  event;  hence  the  chance  that  the  event  will 
happen  in  one  of  these  two  ways  is 

a,  be.  -f-  a^  b,  a,       a^ 

■      \  ,   -     ,  or  T^  +  X  • 
bi  b^  b^       b^ 

Similar  reasoning  will  apply  to  any  number  of 
exclusive  ways  in  which  an  event  may  happen. 


PROBABILITY.  249 

Hence  if  an  event  can  happen  in  n  ways  which  are 
mutually  exclusive,  and  if  /i,  /.,,  /s,  ••.,  /„  are  the 
probabilities  that  the  event  will  happen  in  these  dif- 
ferent ways  respectively,  the  probability  that  it  will 
happen  in  some  one  of  these  ways  is/i+/2+/3 
+  ...+A. 

Example  i.  Find  the  chance  of  throwing  at  least  .8  in  a 
single  throw  with  two  dice. 

8  can  be  thrown  in  5  ways,  .*.  the  chance  of  throwing    8  =  ^ 

9  can  be  thrown  in  4  ways,  .*.  the  chance  of  throwing  9  =  ^ 
10  can  be  thrown  in  3  ways,  .*.  the  chance  of  throwing  10  =  j\ 
ir  can  be  thrown  in  2  ways,  ••.  the  chance  of  throwing  11  =  3!^ 
12  can  be  thrown  in  i  way,  .-.  the  chance  of  throwing  12  =  ^^. 

These  ways  being  mutually  exclusive,  and  8  at  least  being 
thrown  in  each  case, 

the  required  chance  =  A+8^  +  A  +  A  +  ^  =  tV 

Example  2.  One  purse  contains  2  dollars  and  4  half-dollars, 
a  second  3  dollars  and  5  half-dollars,  a  third  4  dollars  and  2 
half-dollars.  If  a  coin  is  taken  from  one  of  these  purses  selected 
at  random,  find  the  chance  that  it  is  a  dollar. 

The  chance  of  selecting  the  ist  purse  =  \\ 
the  chance  of  then  drawing  a  dollar  =  |  =  ^; 
.-.  the  chance  of  drawing  a  dollar  from  ist  purse  =  ^  •  ^  =  ^. 

Similarly,  the  chance  of  drawing  a  dollar  from  2d  purse  =  \\ 
and  the  chance  of  drawing  a  dollar  from  3d  purse  =  f . 
.-.  The  required  chance  =  \  +  \  +  ^  =  \\- 

It  is  very  important  to  note  that  when,  as  in  the  two  ex- 
amples given  above,  the  probability  of  an  event  is  the  sum  of 
the  probabilities  of  two  or  more  separate  events,  these  separate 
(vents  must  be  mutually  exclusive^ 


2SO  ALGEBRA. 

*  361.  If  p  denote  the  chance  of  an  evefit  happening  in 
one  trial,  and  q  ==  i  _  p ;  then  the  chance  of  its  hap- 
pening r  times  exactly  in  n  trials  is  "Cr  p'  q"~'. 

For  if  we  select  any  particular  set  of  r  trials  out  of 
the  whole  number  n,  the  chance  that  the  event  will 
happen  in  every  one  of  these  r  trials  and  fail  in  the 
rest  is/''/'"''  (§§  355,  357);  and  as  in  the  n  trials 
there  are  "Cr  sets  of  r  trials,  which  are  mutually  ex- 
clusive and  equally  applicable,  the  chance  that  the 
event  will  happen  r  times  exactly  in  n  trials  is 

*  362.  The  chance  that  an  eveiit  will  happen  at  least 
r  times  in  n  trials  is 

/«  +  ''C^r-^q  +  "C.p'^-^g''  +  .-.  +  "C/-^''-^ 

or  the  sum  of  the  first  n  —  r  +  i  terms  of  the  expajision 

^/(p  +  q)"- 

For  an  event  happens  at  least  r  times  in  n  trials,  if 
it  happens  n  times,  or  ;/  —  i  times,  or  n  —  2  times, 
...,  orrtimes;  and  if  in  "Crp''q"~''wQ  put  r  equal 
to  n,  71  —  I,  71  —  2,  .».,  r,  in  succession,  and  add  the 
results,  remembering  that  "C-r  =  "C  we  obtain  the 
expression  given  above. 

Example.  In  5  throws  with  a  single  die,  find,  (i)  the 
chance  of  throwing  exactly  3  aces,  (2)  the  chance  of  throwing 
at  least  3  aces. 


PROBABILITY.  2$  I 

Here p  —  \^  ^  =  f ,  «  =  5,  ^  =  3  ;  hence 
the  chance  of  throwing  exactly  3  aces  =  ^C^  (1)^  (|)2  =  Yy^L. 

The  chance  of  throwing  at  least  3  aces  is  the  sum  of  «  —  r  +  i, 
or  3,  terms  of  the  expansion  of  {\  -\-  \)^^  or 

the  chance  =  {\f  +  5  (i)Ht;  f  10  i\y  (t)^  =  ^A- 

*363.  Expectation.  If  /  be  a  person's  chance  of 
winning  a  sum  of  money  M,  then  Mp  is  called  his 
expectation,  or  the  value  of  his  hope.  The  phrase 
probable  value  is  often  applied  to  things  in  the  same 
way  that  expectation  is  to  persons. 

Example.  A  and  B  take  turns  in  throwing  a  die,  and  he 
who  first  throws  a  6  wins  a  stake  of  $  22.  If  A  throws  first, 
find  their  respective  expectations. 

In  his  first  throw,  A's  chance  is  i  ;  in  his  second  throw,  it 
is  {\y^  X  ^  ;  in  his  third,  it  is  {\y  x  \ ;  and  so  on. 

Hence  A's  chance  =  H^  +  (1)^  +  (6)*  +  •••  1- 

Similarly,  B's  chance  =  I  •  i  { i  +  (1)'-^  +  (|)*  +  ..•}• 

Hence  A's  chance  is  to  B's  as  6  is  to  5  ;  or  their  respective 
chances  are  ^^^  and  ^^. 

Therefore  their  expectations  are  $  12  and  %  10  respectively. 

Note.  The  theory  of  probability  has  its  most  important 
applications  in  Insurance  and  the  calculation  of  Probable  Error 
in  physical  investigations.  It  is  also  applied  to  testimony  and 
causes.  But  the  limits  of  this  treatise  exclude  further  consider- 
ation either  of  the  theory  or  its.  applications.  For  a  fuller 
treatment  the  student  may  consult  Hall  and  Knight's  Higher 
Algebra,  Todhunter's  Algebra,  Whitworth's  "Choice  and 
Chance,"  and  the  articles  Annuities,  Insurance,  and  Proba- 
bility in  the  "Encyclopaedia  Britannica."  A  complete  account 
of  the  origin  and  development  of  the  subject  is  given  in  Tod- 
hunter's "  History  of  the  Theory  of  Probability  from  the  time  of 
Pascal  to  that  of  Laplace." 


252  ALGEBRA. 

EXERCISE  48. 

1.  Find  the  chance  of  throwing  9  at  least  in  a  single 
throw  with  two  dice. 

2.  One  compartment  of  a  purse  contains  3  half-dollars 
and  2  dollars,  and  the  other  2  dollars  and  i  half-dollar.  A 
coin  is  taken  out  of  the  purse ;  show  that  the  chance  of  its 
being  a  dollar  is  y^g. 

3.  If  8  coins  are  tossed,  find  the  chance,  (1)  that  there  will 
be  exactly  3  heads,  (2)  that  there  will  be  at  least  3  heads. 

4.  If  on  an  average  i  vessel  in  every  10  is  wrecked,  find 
the  chance  that  out  of  5  vessels  expected,  (i)  exactly  4  will 
arrive  safely,  (2)  4  at  least  will  arrive  safely. 

5.  If  3  out  of  5  business  men  fail,  find  the  chance  that 
out  of  7  business  men,  (i)  exactly  5  will  fail,  (2)  5  at  least 
will  fail. 

6.  Two  persons,  A  and  B,  engage  in  a  game  in  which 
A's  skill  is  to  B's  as  3  to  4 ;  find  A's  chance  of  winning  at 
least  3  games  out  of  5. 

7.  If  A's  chance  of  winning  a  single  game  against  B  is 
f,  find  the  chance,  (i)  of  his  winning  exactly  3  games  out 
of  4,  (2)  of  his  winning  at  least  3  games  out  of  4. 

8.  A  person  is  allowed  to  draw  two  coins  from  a  bag 
containing  4  dollars  and  4  dimes;  find  the  value  of  his 
expectation. 

9.  From  a  bag  containing  6  dollars,  4  half-dollars,  and  2 
dimes,  a  person  draws  out  3  coins  at  random ;  find  the  value 
of  his  expectation. 


PROBABILITY.  253 

10.  Two  persons  toss  a  dollar  alternately,  on  condition 
that  the  first  who  gets  "heads"  wins  the  dollar;  find  the 
expectation  of  each. 

11.  Find  the  worth  of  a  lottery-ticket  in  a  lottery  of  100 
tickets,  having  4  prizes  of  $  100,  10  of  $  50,  and  20  of  $  5. 

12.  Three  persons,  A,  B,  and  C,  take  turns  in  throwing  a 
die,  and  he  who  first  throws  a  5  wins  a  prize  of  ^  182  ;  show 
that  their  respective  expectations  are  $  72,  $60,  and  $50. 

13.  A  has  3  shares  in  a  lottery  in  which  there  are  3  prizes 
and  6  blanks ;  B  has  i  share  in  a  lottery  in  which  there  is  i 
prize  and  2  blanks.     Compare  their  chances  of  success. 

14.  Show  that  the  chance  of  throwing  more  than  15  in 
one  throw  with  three  dice  is  j^-g. 

15.  Compare  the  chances  of  throwing  4  with  one  die,  8 
with  two  dice,  and  12  with  three  dice. 

16.  There  are  three  events  A,  B,  C,  one  of  which  must, 
and  only  one  can,  happen.  The  odds  are  8  to  3  against  A, 
5  to  2  against  B ;  find  the  odds  against  C. 

17.  A  and  B  throw  with  two  dice  ;  if  A  throws  9,  find  B's 
chance  of  throwing  a  higher  numbsr. 

1 8.  The  letters  in  the  word  Vermont  are  placed  at  random 
in  a  row ;  find  the  chance  that  any  two  given  letters,  as  the 
two  vowels,  are  togetlier. 


254  ALGEBRA. 


CHAPTER   XX. 
CONTINUED    FRACTIONS. 

364.    An  expression  of  the  general  form 
b 


a  + 


d 


is  called  a  Continued  Fraction. 

We  shall  consider  only  the  simple  form 


«i  + 


«2  +  ^+..., 

3 


a^ 


in  which  a^y  a^,  a^,  ...  are  positive  integers. 

This  is  often  written  in  the  more  compact  form 

III 

a  H — . . . 

The  quantities  a^,  ^2»  ^3»  •  •  •  ^^e  called  quotients,  or 
partial  quotients.  A  continued  fraction  is  said  to  be 
terminating  or  non- terminating,  according  as  the 
number  of  the  quotients  a^,  a^,  a^,  ...  is  limited  or 
unlimited.  Any  terminating  continued  fraction  can 
evidently  be  reduced  to  an  ordinary  fraction  by  sim- 
plifying the  fractions  in  succession,  beginning  from 


CONTINUED   FRACTIONS.  255 

the  lowest.     Hence  any  terminating  continued  frac- 
tion is  a  commensurable  quantity. 

I  a^  a^  (^^1^2+  0  +  ^1 


Thus,      rt^i  + —  =  ^1  + 


a^a^^-  I  a^a^-\-  I 


365.    To   convert  a  given  fraction  into  a  continued 

fraction. 

ffi 
Let  —  be  the  given  fraction ;   divide  m  by  ;/,  and 

let  ^1  be  the  integral  quotient  and  r^  the  remainder; 


///  r,  I 

then  ~  —  a^-\ —  =  a^  -\ — 


Divide  n  by  r^  with  quotient  ^2  and  remainder  ^2, 
then  -  =  «3  +  -^  =  ^2H 

Divide  rx  by  ^2  with  quotient  a^  and  remainder  ^3 ; 
and  so  on. 

TJ  W  I  II 

Hence     —  =  ^.  -| ,    or    a,  -\-  — ; . . . 

i^  .,+  .3+ 

«3  +  --- 

\int<  ;/,  Ui  =  0,  and  a^  is  obtained  by  dividing  n 
by  m. 

The  above  process  is  evidently  the  same  as  that 
of  finding  the  G.  C.  D.  of  in  and  n ;  therefore  if  m 
and  ft  are  commensurable,  we  must  at  length  obtain  0 
as  the  remainder,  and  the  process  terminates. 


256  ALGEBRA. 

Hence  any  fraction  whose  terms  are  commensur- 
able can  be  converted  into  a  terminating  continued 
fraction. 

Example   i.     Reduce  — —  to  a  continued  fraction. 

Here  the  quotients  are  3,  5,  7,  9  ; 

1051  _  III 

329  -3  +  ^  ^  9' 

Example   2.     Reduce  j~  to  a  continued  fraction. 

Here  the  quotients  are  0,  3,  5,  7,  9 ; 

329 I        I        I      I 

1051  -3+    5+   7^   9' 

366.  Convergents.  The  fractions  obtained  by  stop- 
ping at  the  first,  second,  third,  ...,  quotients  of  a 
continued  fraction,  as 

<7  I     '  II 

-i,   «i  +  — ,   ^i  +  — T  -,  ••• 

I  «2  ^2   +      ^3 

or  when  reduced  to  the  common  form, 

a^      a^a^-\-  i      a^  {a^  a^-\-  i)  +  a^ 

1  *  «2        '  ^3  ^2  +  I  '  * " 

are    called    respectively  the  Ji7'st,  secondy  thirds   ..., 
convergents. 

367.  The  successive  convergents  are  alternately  less 
and  greater  than  the  continued  fraction. 

The  first  convergent,  a^ ,  is  too  small,  since  the  part 

...    is    omitted.      The    second   convergent, 

«2  +    «8  + 


CONTINUED    FRACTIONS.  257 


a^-i ,  is  too   great,  for  the   denominator  ^2  is   too 

^2  II. 

small.      The    third    convergent,    a^  -\ ,  is    too 

J  ^2  +  ^8 

small,  for  a^  -\ is  too  great  (^3  being  too  small)  ; 

and  so  on. 


368.    To  establish  the  law  of  formation  of  the  sue- ' 
cessive  convergents. 

If  we  consider  the  first  three  convergents, 

a,  ^    a,a^+  I  ^    a,  (^,  a,  +  i)  +  ^^i  ^      g  ^66. 

we  see  that  the  numerator  of  the  third  convergent 
may  be  obtained  by  multiplying  the  numerator  of 
the  second  convergent  by  the  third  quotient,  and 
adding  the  numerator  of  the  first  convergent;  also 
that  the  denominator  may  be  formed  in  a  similar 
manner  from  the  denominators  of  the  first  two  con- 
vergents. We  proceed  to  show  that  this  law  holds 
for  all  subsequent  convergents. 

Let  the  numerators  of  the  successive  convergents 
be  denoted  by/,, /._,» A^  •••>  and  the  denominators  b\ 
(7p  <72'  ^8»  •••  Assume  that  the  law  holds  for  the  ;/th 
convergent ; 

then  A  =  ^«A-i+A-2,  (i) 

and  ^„  =  a„^„_i  +  ^„_2'  (2) 


258  ALGEBRA. 

The  (;/  +  i)th  convergent  evidently  differs   from 


the  nth.  only  in  having  a,  •] ^  -  in  the  place  of  a„; 

^n  +  I 

hence 

^  an+l  {Cinpn-X   ^Pn-^    ^-pn-1 
an+x{cinqn-\  +   ^„  -  2)    +   Qn-X 

^n  +  lPn   +  Pn~\        ,  ,    x  ,     ,     x 

"■n  +  1  y «   i^   y «  -  1 

Hence  the  law  holds  for  the  {n  +  i)th  convergent, 
if  it  holds  for  the  ;/th.  But  it  does  hold  for  the  third  ; 
hence  it  holds  for  the  fourth ;   and  so  on. 

Therefore  it  holds  universally  after  the  second. 

The  method  of  proof  employed  in  this  article  is 
known  as  Mathematical  Induction. 

Example    i.     Calculate  the  successive  convergents  of 
I        I        I         I       I 

^"'"BT  r+  T+  II  +  2" 


Here 

^1,    a^,    ^3,    ^4, 

^5^      ^6J 

are 

2,      6,      I,      I, 

II,      2. 

Hence  ^^  = 
9i 

2 
I  ' 

A_2  ,  1-L3. 

A 

I  X  13  +  2      15 
1x6+1   ~  7  ' 

A     IX  15 +  13 

28 

Qz 

^4       1x7  +  6 

13' 

A 

11x28+15      323 

A     2  x  323  +  28 

674 

^6 

II  X  13  +  7  ""  150' 

^6         2X150+13 

313 

CONTINUED   FRACTIONS.  259 

Example   2.     Find  the  successive  convergents  of 

I        I        I        I        I        I      I 

2T  2T  3+   i~+  4T  2T  6* 

Here  a^,    a^,    n^,    a^,    ^5,    ^5,    ^7,    a^, 

are  0,      2,      2,    3,      i,      4,      2,      6. 

wpnr.     9      i      ^      _!       9       _43        95       ^ 
nence     j,     3,     ^,     17,     22'     105'     232'     1497* 

are  the  successive  convergents. 


EXERCISE  49. 

Compute  the  successive  convergents  to 
I       I       I       I       II 

I     I     I      I     I 

^'  7+7+6  +  ^10* 

I      1     I     I     I    I 

Express  each  of  the  following  fractions  as  a  continued 
fraction,  and  find  its  convergents  : 

Reduce  to  a  continued  fraction,  and  find  the  fourth  con- 
vergent to,  each  of  the  following  numbers  : 

12.    0.37.       13.    1. 139.  14.   0.3029.         15.    4.316. 

Write  0.37  as  a  common   fraction,  ^^\   then  proceed  as 
above. 


260  ALGEBRA. 

369.  The  difference  betiveen  any  tivo  consecutive  con- 
vcr gents  is  iDiity  divided  by  the  product  of  their  de- 
nofninaiors  ;  that  is^ 

Vu  ^H  +  1  ^n^>i+l 

The  law  holds  for  the  first  two  convergents,  since 

^1      ai  a^  -\-  I        I 


^2 


(I) 


Assume  that  the  law  holds  for  — — ^  and  — ,  so  that 

qn-x        qn 

Pnqn-l--qnPn-l=    Ij  (2) 

then  by  §  368 

Pn  qn+1  '^Pn+l  qn  =  Pn  (^«+l  ^«  +  ^..-l)  ^  qn  {^n+lPn  +  Pn-l)       ' 
=  Pnqn-l-qnPn-l 

=  I,  by  (2). 

Hence,  if  the  law  holds  for  one  pair  of  consecutive 
convergents,  it  holds  for  the  next  pair.  But  by  (i), 
the  law  does  hold  for  \hQ first  pair;  therefore  it  holds 
for  the  second  pair ;    and  so  on. 

Therefore  it  is  universally  true. 

370.  Any  convergent/„-r  ^„  is  in  its  lowest  terms. 
For  if /„  and  q„  had  a  common  factor,  it  would  also 
be  a  factor  o{  p,,q„j^^  ^ p„^^q„,  or  unity;  which  is 
impossible. 

*  The  expression  x  ~j denotes  "the  difference  between  ;r  and ^." 


CONTINUED    FRACTIONS.  26 1 

371.  Let  X  denote  the  value  of  any  continued  frac- 
tion ;  then,  by  §  367,  x  Hes  between  any  two  consecu- 
tive convergents;   hence 

,'.    x^^<-^—  <-^ .    §§369,368. 

qn  qnqn  +  l  (^«)^«  +  l 

P  T 

That  is,  —  differs  from  x  by  less  than 


qn  {q.y'<h.  +  . 

Hence  the  «th  convergent  is  a  near  approximation 
when  q„  and  a„  +  ^  are  large. 

372.  Eac/i  convergent  is  nearer  to  the  continued 
fraction  than  any  of  the  preceding  convergents. 

Let  x  denote  the  continued  fraction,  and  A  the 
complete   (ft  +  2)th  quotient,  a„  +  ^  -\ ... ;    then 

A  +  3  ^"+3   + 

X  differs  from  only  in  having  A  in  the  place  of 

a„^^\  hence 


^  qn  +  l  +  qn 


.'.    X 


Pn  _  A{P»  +  \qn--pnqn+\)    _ 


q„  q„  (^  ^„  + 1  +  ^„)  q,,  {A  ^„  + 1  +  q,)  ' 

and 

Pn+\  ^^_p„Ar\q>,~  P>,q..+\  _  I 


qn  +  1  q„^.  i{A  q„  + 1+  q,i)       q,,  + 1  (^  $^„  + 1  +  q,^  ' 

Now  ^  >  I,  and  q„  <  ^„  +  ,;    hence  the  difference 
between  the  («  +  i)th  convergent  and  x  is  less  than 


262  ALGEBRA. 

the  difference  between  the  ;/th  convergent  and  x ; 
that  is,  any  convergent  is  nearer  to  the  continued 
fraction  than  the  next  preceding  convergent,  and 
therefore  than  any  preceding  convergent. 

From  this  property  and  that  of  §  367,  it  follows 
that 

The  convergent s  of  an  odd  oi^der  continually  increase ^ 
but  are  always  less  than  the  continued  fraction. 

The  convergents  of  an  even  order  continually  decrease^ 
but  are  always  greater  than  the  continued  fraction. 

Example.    Find  the  successive  convergents  to  3. 141 59. 

Here  the  quotients  are     3,    7,     15,      i,     25,     i,    7,     •..; 
hence  the  convergents  are   f ,  ^,  ||f ,  f  f |,  •  •  • 

If  the  4th  convergent,  which  is  greater  than  3. 141 59,  be  taken 
as  its  value,  the  error  will  be  less  than  i  ^  25  (113)^  and  there- 
fore less  than  i  -^  25  (loo)^  or  0.000004. 

The  convergents  above  will  be  recognized  as  the  approxi- 
mate values  of  tt,  or  the  ratio  of  the  circumference  of  a  circle  to 
its  diameter.  This  example  illustrates  the  use  of  the  properties 
of  continued  fractions  in  approximating  to  the  values  of  incom- 
mensurable ratios  or  those  represented  by  large  numbers. 

373.  Any  convergent  approaches  more  nearly  the 
value  of  the  continued  fraction,  x,  than  any  other 
fraction  whose  denominator  is  less  than  that  of  the 
convergent. 

f  p^^ 

For  let  the  fraction  -  be  nearer  to  x  than  —  ;  then 
s  q„ 

is  it  nearer  to  x  than  the  (;/  —  i  )th  convergent 
(§  372)  ;   and  since  x  lies  between  the  ;/th  and  the 


CONTINUED    FRACTIONS.  263 

(;;_i)th    convergent,   r -^  s   does   also;    hence  we 
have 

s     q„-\     qn     qn-\        q»qn-i 

s  ,   . 

.'.     rq„_i^sp„_i  <-.  (I) 

Now  the  first  member  of(i)  is  an  integer;  hence 

r  pn 

s  >  q„;    that  is,  if  -  is  nearer  x  than  is  — ,  J  >  q„. 


374.  Periodic  Continued  Fractions.  A  continued 
fraction  in  which  the  quotients  recur  is  called  a 
periodic^  or  recurring^  continued  fraction. 

Any  quadratic  surd  can  be  expressed  as  a  periodic 
continued  fraction.  We  give  the  following  example 
to  illustrate  this  principle,  and  to  exhibit  the  use  of 
the  properties  of  continued  fractions  in  approximating 
to  the  value  of  a  quadratic  surd. 

Example.  Convert  /y/TJ  into  a  continued  fraction,  and  find 
its  convergents. 

Since  3  is  the  greatest  integer  in  ^JTs,  we  write 


*  A/  I 


V15  +  3 


(I) 


_Vl5±3^i  +  \/i5ii3^,+ 


(2) 


6  '6  V^+3' 

VTI+3^6^Vri-3^^^        6_^ 

The  last  fraction  in  (3)  is  the  same  as  that  in  (i);  hence  after 
this  the  quotients  i,  6,  will  recur. 


264  ALGEBRA. 


I        I        I        I 


Hence         v^=3  +  —  gr;  7+  6 


The  quotient  in  each  of  the  identities,  (i),  (2),  (3),  is  the 
greatest  integer  in  the  value  of  its  first  member.  The  nu- 
merator of  each  fraction  is  rationahzed  so  that  the  inverted 
fraction  will  have  a  rational  denominator. 

To  find  the  convergents,  we  have  the  quotients 

3,     I,     6,     I,      6,       I,     6,  ...; 

.3         4       2  7      .31.      i\3      2  4JL      ... 
1>       U       T )      8  '      55 '      63  » 

are  the  first  six  convergents. 

The  error  in  taking  the  sixth  convergent  as  the  value  of 
/v/iy  is  less  than  1-^-6  (63)'^,  and  therefore  less  than  0.0C005. 

375.  Every  periodic  continued  fraction  is  equal  to 
one  of  the  surd  roots  of  a  quadratic  equation  with 
rational  coefficients.  The  following  example  will 
illustrate  this  general  truth. 

I  T  I  I  , 

Example.    Express  i  +  —  -—   — ^   —  •  •  •  as  a  surd. 

Let  X  denote  the  value  of  the  continued  fraction ;  then 
I        I        I        I 


~2  + 

3+   2  + 

3  + 

I 

I 

~2  + 

3  +  (^- 

0' 

ix'^A-ix-T- 

=  0. 

The  continued  fraction,  being  positive,  is  equal  to  the  positive 
root  of  this  equation,  or  \  {^^J ^S  ~  i)- 


CONTINUED   FRACTIONS.  265 

EXERCISE   50. 

Reduce  to  a  continued  fraction,  and  find  the  sixth  con- 
vergent to,  each  of  the  following  surds  : 

I.    Vl'  3-    V6-  5-    Vm-  7-   3^5- 


2. 


V^.  4.    \/T^.  6.    V22'  8.   4V10. 


9.  In  each  of  the  above  examples  the  difference  between 
the  surd  and  the  sixth  convergent  is  less  than  what? 

10.  Find  the  first  convergent  to 
I       I       I       I        I 


1  + 


3  +  5  +  7  +  9  +  11  + 
which  differs  from  it  by  less  than  0.000 1. 

11.  Find  the  first  convergent  to  VToi  that  differs  from  it 
by  less  than  0.0000004. 

12.  Given  that  a  metre  is  equal  to  1.0936  yards,  show 
that  the  error  in  taking  222  yards  as  equivalent  to  203  metres 
will  be  less  than  0.000005. 

13.  A  kilometre  is  very  nearly  equal  to  0.62138  miles ;  show 
that  the  error  in  taking  103  kilometres  as  equivalent  to  64 
miles  will  be  less  than  0.000025. 

14.  Find  the  first  six  convergents  to  the  ratio  of  a  diago- 
nal to  a  side  of  a  square.  The  difference  between  each  of 
the  six  convergents  and  the  true  ratio  is  less  than  what? 

15.  Express  3  +  ^  A:  ^  •  •  •  as  a  surd. 

16.  Express  — ; : ...  as  a  surd. 

I  +  3  +  I  +  3  + 


266  ALGEBRA. 


CHAPTER   XXI. 
THEORY   OF   EQUATIONS. 

376.  The  General  Equation.  Let  n  be  any  positive 
integer,  and/i,/2.A,  •••>/«>  be  any  rational  known 
quantities;  then  the  equation 

X''  +A-^"~'  +A-^""'  +  •••  +A-i-^  +  A  =  0     (A) 

will  be  the  general  type  of  a  rational  integral  equa- 
tion of  the  ?/th  degree.     In  this  chapter  we  shall  let 

F{x)  =  x''+  A  -^'^-^  +  A  ^"~'  +  •  •  •  +  A 
and  write  equation  (A)  briefly  F  (x^  =  0.* 

377.  A  Root  of  the  equation  F{x)  =  0  is  any  value 
of  ;r,  real  or  imaginary,  that  causes  the  function,  F{x)y 
to  vanish. 

378.  Reduction  to  the  form  F(^x)  =  0.  In  general, 
any  equation  in  x  having  rational  coefficients  can  be 
transformed  into  an  equation  of  the  form  F(x')  =  0. 
The  following  example  will  illustrate  the  general 
truth. 

*  What  properties  of  the  equation  F{x)=0  belong  also  to  the 
equation  formed  by  putting  any  rational  integral  function  of  x  equal 
to  zero,  the  reader  will  readily  discover. 


THEORY  OF  EQUATIONS.  267 

I  —  x^      x—^  4-  3 

Example.    Reduce  — — —  =  — r to  the  form  oiF{x)  —  0. 

i  +  -^        X^+2. 

Clearing  the  given  equation  of  fractions  we  obtain 

x^  —  x^  +  2  —  2  x^  =  x~'^  +  ^  +  I  +  Z^' 
Multiplying  by  x  to  free  of  negative  exponents,  we  obtain 

X^  —  X^    -2X^  =  I  +  2X+  :iX^.  (l) 

To  transform  (i)  into  another  equation  with  integral  expo- 
nents, put  x=jy\  6  being  the  L.  C  M.  of  the  denominators  of 
the  fractional  exponents  of  x.     We  thus  obtain 
y  _^i8  _  2jio  =  I  +  2j/«  +  3>/i2, 
or  ys  +  3^2  4.  2  jlO  _y  -I-  2_;/6  +  I  =  0,  (2) 

which  is  in  the  required  form. 

The  roots  of  (i)  and  (2)  hold  the  relation  x=j^^. 

EXERCISE  51. 

Reduce  the  following  equations  to  the  form  I^(x)  =  0 : 

2  L 

I. 3^.-  +  ^x-  —  1  =  1. 


2. 

x^- 

-  I 

I 

I 

^-2 

I  + 

J 

-3-* 

3- 

Vi 

-X^: 

4.     V2  X  —  ■zx^  —  X  =  \/l  —  X. 

379.   Divisibility  of  F(x).     If  F  (x)  is  divided  b), 
X  —  a,  the  remainder  will  be  F  (a). 


268  ALGEBRA. 

Divide  Fix)  hy  x  —  a  until  a  remainder  is  obtained 
that  does  not  involve  x.  Let  /\  (;r)  denote  the 
quotient,  and  R  the  remainder;   then 

F{x)  =  (x-~a)  F,  {x)  +  F. 

Since  R  does  not  involve  x,  it  is  the  same  for  all 
values  of  x.     Putting  ;r  =  ^,  v^e  obtain 

F{a)  =  0  X  ^1 W  +  ^  =  ^,  the  remainder. 

//*  F  (a)  =  0,  F  (x)  is  divisible  by  x  —  a ;  ajid  con- 
versely. 

Example.     If  n  is  even,  show  that  x"  —  ^"  is  divisible  by 

Since  n  is  even,  and  F{x)  m  x"  —  b^; 

.-.  Fi-b)^b^-b^  =  0. 
Hence  x»  —  b"  is  divisible  by  ;ir  —  (—  b),  or  x  +  b. 

380.  If  3.  is  a  root  of  the  equation  F  (x)  =  0,  that  is, 
if  F  (a)  =  0,  theft  F  (x)  is  divisible  by  x  —  a  (§  379). 
Conversely,  if  F  (x)  is  divisible  by  x  —  a,  the  71 
F  {a)  =  0,  that  is,  a  is  a  root  of  the  eqtmtion  F  (x)  =  0. 

381.  Horner's  Method  of  Synthetic  Division. 
Let  it  be  required  to  divide 

Ax^-\-  Bx'^+  Cx^  D  hy  x-a. 

In  the  usual  method  given  below,  for  convenience  we  write 
the  divisor  to  the  right  of  the  dividend  and  the  quotient  below 
it. 


+  *{Aa^  +  Ba  +  C) 


THEORY  OF  EQUATIONS.  269 

Ax^+Bx^  +Cx         +D 

*Ax^-  Aax^ 
(Aa  +  B)x^ 
*(Aa  +  B)x^  -{Aa'^  +  Ba)x 
iAa^-\-Ba+C)x 
*  (A  a'^  +  B a  ^  C)  X -  (A  a^  -{-  B a^+  C a) 

Aa^  +  Ba^+Ca  +  D 

Here  the  remainder,  A  a^  -{-  B  a^  -{-  Ca  +  D,  is  the  value  of 
the  dividend,  A  x^  +  B  x^  +  C  x  -{-£>,  ior  x  =  a,  which  aliorcis 
a  second  proof  of  §  379. 

In  the  shorter  or  synthetic  method,  we  write  the  coefficients 
of  the  dividend  with  a  at  their  right  as  below : 


^B  +C  +B  \a_ 


+  Aa         +Aa^-{-Ba  +Aa^-]-Ba^-\-Ca 


Aa  +  B  Aa^  +  Ba  +  C        Aa^  +  Ba'^+Ca  +  D 

Multiplying  A  by  a,  writing  the  product  under  B^  and  adding, 
we  obtain  Aa  +  B.  Multiplying  this  sum  by  «,  writing  the 
product  under  C,  and  addin<i^,  we  obtain  A  a'^  +  B  a  +  C.  In 
like  manner  the  last  sum  is  obtained. 

Now  A  and  the  first  two  sums  are  respectively  the  coefficients 
of  x%  X,  ana  x^  in  the  quotient  obtained  above  by  the  ordinary 
method,  and  the  last  sum  is  the  remainder. 

In  like  manner  any  rational  integral  function  of  x  may  be 
divided  by  :r  —  ^.  If  any  power  of  x  is  missing,  its  coefficient 
is  zero,  and  must  be  written  in  its  place  with  the  others. 

The  shorter  or  synthetic  method  of  division  is 
obtained  from  the  usual  method  given  above  by- 
omitting  the  powers  of  x  and  the  terms  marked  with 
an  asterisk  (*),  by  changing  the  minus  signs  to  plus 
(which  is  the  same  thing  as  changing  the  sign  of  the 


270  ALGEBRA. 

second  term  of  the  divisor),  and  then  adding  instead 
of  subtracting. 

Example  i.   Divide  x*  +  x^  -  2gx^  -  gx  +  180  by  :f-  6. 
Write  the  coefficients  with  6  at  their  right  and  proceed  as  below. 

I        +1         -  29        -    9        +180        [6^ 
+  6        +42        +78        +414 


I        +7        +13        +69        +594 

Thus  the  quotient  =  x^  -{-  y  x^  -j-  i^x  +  6g, 
and      the  remainder  =  7^(6)  =  594. 


Example  2.    Divi 

ide  x^  +  x^  —  2gx^  —  gx-\ 

^-(-6). 

I        +  I 

—  29        —    9        +  180 

-6 

+  30        -    6        +90 

I        -5 

+    1         -  15        +270 

Here  the  quotient 

-^-5^'  +  ^- 15, 

gx+  180  by  :r  +  6, 
t.6 


and      the  remainder  =  7^(—  6)  —  270. 

Example  3.    Divide  x^  +  21  x  ■]-  342  by  or  +  6. 

I        +0        +21        +342        |-6 
-  6        +36        -  342 


1-6+57  0 

Here  the  quotient  =  x"^  —  6x  +  57, 
and      the  remainder  =  i^(— 6)  =  0. 

Hence  the  division  is  exact,  and  —  6  is  a  root  of  F{x)  =  0. 

382.  When  one  root  of  an  equation  is  known,  the 
equation  may  be  depressed  into  another  of  the  next 
lower  degree,  the  roots  of  which  are  the  remaining 
roots  of  the  given  equation. 


THEORY   OF   EQUATIONS.   .  27 1 

Example.  Solve  x^  -  12  ;ir-  +  45  ;r  —  50  =  0,  one  root 
being  5. 

One  root  being  5,  one  factor  of  F(x)  \s  x  —  $   (§  380)-     % 

division  the  other  factor  of  /^(x)  is  found  to  be  x'^  —  7 x+  10. 

Hence  by  §  137  the  two  roots  required  are  those  of  the  quadratic 

equation 

x^-7x-{-  10  =  0.  (I) 

The  roots  of  (i)  are  evidently  5  and  2. 

EXERCISE  52. 

Sy  §  379)  show  that 

1.  When  n  is  integral,  x"  —  a"  is  divisible  by  :t:  —  ^. 

2.  When  //  is  odd,  x"  +  a"  is  divisible  hy  x  -{-  a. 

3.  When  n  is  even,  x"  +  a"  is  «<?/  divisible  by  either  x  —  a 
or  X  +  a. 

By  Horner's  method  divide 

5.  2  x^  -{-  /^x^  —  x^  —  16  X  —  i2byjc4-4;bya:  +  3« 

6.  3  ^*  —  27  x^  -\-  14  ^  +  1 20  by  .:v  —  6  ;  by  jf  +  5. 

7.  Evaluate  2  .;c*  —  3  j;^  +  3  jtr  —  i  for  jt:  =  4,  jc  =  —  3, 
^  =  3. 

8.  One  root  of  jv^  —  6  a^^  +  10  .r  —  8  =  0  is  4  ;  find  the 
others. 

9.  One  root  of  x^  -{-  S  x^  -{-  20  Ji;  +  16  =  0  is  —  2  ;  find 
the  others. 


272  ALGEBRA. 

10.  One  root  o(  x^  -\-  2  a^  —  23  x  —  60  =  0  is  —  3  ;  find 
the  others. 

1 1.  One  root  of  ^^  —  7  ::c^  +  36  =  0  is  —  2  ;  find  the 
others. 

12.  Two  roots  of  a'*  -\-  x^  —  29  :r^  —  9  "^  +  180  =  0  are 
3  and  —  3  ;  find  the  others. 

13.  Two  roots  of  a;*  —  4;^^  —  8  :^  +  32  ==  0  are  2  and  4 ; 
find  the  others. 

14.  If  the  coefficients  of  J^(x)  are  all  positive,  the  equa- 
tion jF{x)  —  0  can  have  no  positive  root. 

15.  If  the  sum  of  the  coefficients  of  the  even  powers  o(  x 
in  I^(x)  is  equal  to  the  sum  of  those  of  the  odd  powers,  one 
root  of  the  equation  J^  (x)  =  0  is  —  i. 

383.  Eveiy  equation  of  the  form  F  (x)  =  0  has  a 
rooty  real  or  imaginary. 

For  the  proof  of  this  theorem  see  Burnside  and  Panton's 
or  Todhunter's  "  Theory  of  Equations."  The  proof  is  too  long 
and  difficult  to  be  given  here, 

384.  Number  of  Roots.  Every  equation  of  the  nth 
degree  has  n,  aitd  only  n,  roots. 

By  §  383,  the  equation  F{x^  =  0  has  a  root.  Let 
^,  denote  this  root;  then,  by  §  380,  F(x)  is  divisible 
by  X  —  aj,,  so  that 

E(ix)^(x-a,)E,(x\  (i) 


THEORY   OF  EQUATIONS.  2/3 

in  which  F^  (x)  has  the  form  of  F(x)y  and  is  of  the 
(;/  —  i)th  degree.  Now  the  equation  F^  (x)  =  0  has 
u  root.     Denote  this  root  by  a.^;   then 

F,(x)  =  (x-a,)F,(x),  (2) 

in  which  F^  (x)  is  of  the  (n  —  2)th  degree. 

Repeating  this  process  7i—  i  times  we  finally  obtain 

F,,.^{x)=x-a,,  (n) 

From  (i),  (2),  ...,  (n),  we  obtain 
F(x)  =  (x  -  a,)  F,  (x) 

=  (x-  a^)  (x  -  a^)  F,  (x) 

=  (x  —  a^  (x  —  a^  (x  —  a^  ...  {x  -  a„).      (3) 

Now,  since  F(x)  vanishes  when  x  has  any  one  of 
the  values  a^,  ^2.  ^3»  •••.  ^..^  and  only  then,  the  equa- 
tion F(x^  =  0  has  7iy  and  only  ;/,  roots. 

385.  Equal  Roots.  If  two  or  more  of  the  factors 
X  —  au  AT  —  ^2.  •••»  ^  —  (^n  are  equal,  the  equation 
/^(,r)  =  0  is  considered  as  having  two  or  more 
equal  roots. 

Thus,  of  the  equation  (.r  -  4)8 ("^  +  5)^  (-^  -  7)  =  O5  three 
roots  are  4  each,  and  two  are  —  5  each. 

386.  Equation  having  Given  Roots.  An  equation 
can  be  formed  of  which  the  roots  shall  be  any  given 


274  ALGEBRA. 

quantities,  by  subtracting  each  of  these  quantities 
from  ;r,  and  putting  the  product  of  the  resuhs  equal 
to  zero. 

Thus,  the  equation  whose  roots  are  2,  3,  and  —  i  is 

(:r  -  2)  (;r  -  3)  (;ir  +  i)  =  0,    or   x^  -  4x'^  +  x  +  6  -  0. 

387.    Relations  between  Coefficients   and  Roots. 
In  the  equation 

—  pi  =  the  sum  of  the  roots  ; 

P2  =  the  sum  of  the  prodtLcts  of  the  roots  taken 
two  at  a  time ; 

—  P3  =  the  sum  of  the  products  of  the  roots  taken 

three  at  a  time. 

(—  i)"  Pn  =  the  product  of  the  n  roots. 

If  ^1,  a<i,  rt-^,  ...,  a„  denote  the  n  roots  of  (A),  by 
(3)  of  §  384  we  have  the  identity 

^«  +  A>^"~^  +  •••  +  A  =  (^  —  ^1)  («^  —  ^2)  •••  {^  —  O- 

Multiplying  together  the  factors  of  the  second 
member,  and  equating  the  coefficients  of  like  powers 
of  X  (§  263),  we  obtain  the  theorem. 

Thus,  when  «  =  2,  we  have 

x^  +PiX  +p^=x^  —  (a^  +  a^)  X  +  a^  a^ ; 
r.   —A  =  ^h  +  ^2>  A  =  ^1^2* 


THEORY   OF  EQUATIONS.  275 

When  n  =  3,  we  have 

=  x^—  (a^  +  ^2  +  ^3)  ^^  +  (^1  ^2  +  «i  ^3  +  ^2  ^3)  -^  —  ^1  ^2  ^3  -' 

•••     —A  =  ^1+^2+^3?    A  =  «1^2+^1^3+^2«3>    — A  =  ^1^2^3- 

From  the  laws  of  multiplication  it  is  evident  that 
the  same  relation  holds  when  ;/  =  4,  5,  6,  ... 

If  the  term  in  x"~^  is  wanting,  the  sum  of  the  roots 
is  0,  and  if  the  known  term  is  wanting,  at  least  one 
root  is  0. 

Thus,  in  the  equation  x^  -\-  6  x"^  —  11  x  —  6  =  0,  the  sum  of 
the  roots  is  0;  tiie  sum  of  their  products  taken  two  at  a  time 
is  6;  the  sum  of  their  products  taken  three  at  a  time  is  11 ;  and 
their  product  is  —  6.     Note  that  -  A  =  (~0A'  A  =  (-  ^fPv  •  •  • 

Note.  The  coefficients  in  any  equ  ition  are  functions  of  the 
roots;  and  conversely,  the  roots  are  functions  of  the  coeffi- 
cients. The  roots  of  a  literal  quadratic  equation  have  been  ex- 
pressed in  terms  of  the  coefficients  (§  144).  The  roots  of  a 
literal  cubic  or  biquadratic  equation  may  also  be  expressed 
in  terms  of  the  coefficients,  as  will  be  shown  in  §§  421,  423. 
But  the  roots  of  a  literal  equation  of  the  fifth  or  higher  degree 
cannot  be  so  expressed,  as  was  proved  by  Abel  in  1825. 

EXERCISE  53. 

By  §  3S6,  form  the  equations  whose  roots  are  given  below, 
and  verify  each  equation  by  §  387  : 

I-    I,  -  3,  -  5-  4-   -  §,  3  ±  V^,  5- 

2.  I,   V2,  —  a/2.  5.      I,   2,   V'3- 

3.  I,  ±  A/3,  ±  V5.  6.   3,  -  4,  V^^. 


2/6  ALGEBRA. 


7.   i,  I  ±  V3,  I  ±  V5-  9-    V3»  V^-2- 


8.    ±  V-  I,  3  ±  V-2,  2. 

Note.  In  each  of  the  above  examples,  the  student  sTiould 
note  that  the  coefficients  of  the  equation  obtained  are  a// 
rational  whenever  the  surd  or  imaginary  roots  occur  in  conju- 
gate pairs. 

10.  Solve  4x^ —  24.x^  -{-  2^x  -{-  18  =  0,  having  given 
that  its  roots  are  in  arithmetical  progression. 

Reduce  the  equation  to  the  form  ^{x)  =  0,  and  denote  its 
roots    by  a  —  d,   a,    and   a  -{-  b ;  then  by  §  387 

3  a  :=  6,  3  «2  _  ^2  ^  .2_3^  a  (^2  _  ^2>)  ^   _  9.  (i) 

Hence  ^  =  2,  and  b  =  ±  ^\  therefore  the  roots  are  —  \,  2, 
|.  The  values  of  a  and  b  must  satisfy  all  three  of  the  equations 
in  (I). 

11.  Solve  4  ;c^  +  16  ^^  —  9  ^  —  36  =  0,  the  sum  of  two 
of  the  roots  being  zero. 

12.  Solve  ^x^  -{-  20  jc^  —  23  a:  +  6  =  0,  two  of  the  roots 
being  equal. 

13.  Solve  3  ..^  —  26  Ji:^  +  52  ^  —  24  =  0,  the  roots  being 
in  geometrical  progression. 

388.  Imaginary  Roots  /;/  the  equation  F  (x)  =  0, 
imaginary  roots  oeeur  in  conjugate  pairs  ;  that  is,  if 
a  +  hV—  lisa  root  of  F  (x)  =  0,  then  a  —  b  V  —  I 
is  also  a  root. 


THEORY   OF   EQUATIONS.  2/7 


If  <2  +  3  V  —  T  be  substituted  for  x  \\\  F(x),  all  its 
terms  will  be  real  except  those  containing  odd  powers 
of  ^  V  —  I,  which  will  be  imaginary.  Representing 
the  sum  of  all  the  real  terms  by  A,  and  the  sum  of 
all  the  imaginary  terms  by  ^  V  —  i,  we  have 


F{a  +  b^-l)=A  +  B^/-l  =  0.       (i) 


Now  F(a  —  b  V  —  i)  will  evidently  differ  from 
F{a  -\-  b  V  —  i)  only  in  the  signs  of  the  terms 
containing  the  odd  powers  of  b  V  —  I  ;  that  is,  in  the 
sign  of  ^  V  —  I ;   hence 


F{a-b^/ -  i)  =A-B\^^^, 
From  (i)  by  §  124,  A  =  0,  and  ^  =  0;  hence 
F{a  -  b  V^)  =  0. 

Example.  Onerootof  ji-8  — 4:ir2  +  4;ir  — 3  =0  is  K^+a/"^)' 
find  the  others. 

Since  ^(i  +  /y/—  3)  is  one  root,  ^i  —  \/^~3)  is  a  second  root. 
The  sum  of  tliese  two  roots  is  i,  and  1)y  §  387  the  sum  of  all 
three  roots  is  4;  hence  the  third  root  must  be  3. 

Here  F{x^  ^  (.r  -  3)  (^  -  ^  -  i  V^)  (^  -  *  +  ^  \/=l) 

=  (^-3)[(^-|)*+|]  =  (^-3)(^^-^+i); 

that  is,  the  real  factors  of  ;t-8  —  4  ;i-2  +  4  ;r  —  3  are  ^  —  3  and 

389.  By  §  388  an  equation  of  an  odd  degree  must 
have  at  least  one  real  root;  while  an  equation  of  an 
even  degree  may  not  have  any  real  root. 


278  ALGEBRA. 

390.  Real  Factors  of  F{x).  Since  {x~a  —  b^/~  i) 
(x  —  a  -\-  b  ^/  —  i')  =  (^x  —  a^  +  b'^,  the  imaginary 
factors  oi  F  {£)  occur  in  conjugate  pairs  whose  pro- 
ducts are  of  the  form  {x  —  a)'^  +  b'^. 

Hence,  F  (x)  cafi  be  resolved  hito  real  linear  or 
quadratic  factors  in  x. 

391.  To  transform  an  equation  into  another  whose 
roots  shall  be  some  multiple  of  those  of  the  first. 

If  in  the  equation 

^«  +A^"~'  +  A^""'  +Pz^"~'  +  •••  +  A  =  0,    (i) 
we  put  X  =  x^^  ay  and  multiply  by  a'\  we  obtain 

a:/'+ A  ^  ^i  - '+ A  ^'•^i"  ~  '+ A  ^^^1  "'+•••  +/«  a''  =  0.     (2) 

Since  x^  =  ax,  the  roots  of  (2)  are  a  times  those 
of  (I). 

Hence,  to  effect  the  required  transformation,  mtil- 
tiply  the  second  term  of  F  (x)  =  0  ^  the  givett  factor, 
the  third  by  its  square,  and  so  on. 

Any  missing  power  of  x  must  be  written  with  zero 
as  its  coefficient  before  the  rule  is  applied. 

The  chief  use  of  this  transformation  is  to  clear  an 
equation  of  fractional  coefficients. 

Example.  Transform  the  equation  x^  —  ^x^^  -\-  \x  —  ^^  =  0 
into  another  with  integral  coefficients. 


THEORY   OF  EQUATIONS.  279 

Multiplying  the  second  term  by  a,  the  third  by  a^^  the  fourth 
by  a^-,  we  obtain 

x^-lax'^  +  \a'^x- {^a^  =  0.  (i) 

By  inspection  we  discover  that  4  is  the  least  value  oi  a  that  will 
render  the  coefficients  of  (i)  integral.     Putting  «  =  4,  we  have 

;f3  -  10  ;r2  +  28  :ir  -  12  =  0  (2; 

as  the  equation  required. 

The  roots  of  (2)  each  divided  by  4  are  the  roots  of  the  givei 
equation. 

EXERCISE  54. 

1.  One  root  of  ^^—  6x'  -\-  57.^—196  =  0  is  i— 4V— 3: 
find  the  others. 

2.  One  root  of  a;^  —  6  ;c  +  9  =  0  is  |  (3  +  V  —  3)  ;  find 
the  others. 

3.  Two  roots  oi  x^  —  x^  -^  x^  —  x^  +  X  —  1  =  0  are 
—  V^^  and  ^  (i  +  V  —  3)  ;  find  the  others. 

4.  One  root  o{ x^  —  2x^  ■{■  2x—  i=0  \s  ^{i  -\-  -v/  — 3)  ; 
find  the  real  factors  oi  x^  —  2  x^  -\-  2  x  —  i.  Find  the  real 
factors  of  F  {x)  in  the  Examples  from  i  to  3. 

5.  Prove  that,  if  ^t  +  V^  is  a  root  of  F{x)  =  0,  a  —  ^/b 
is  a  root  also.     (See  proof  in  §  388.) 

Transform  the  following  equations  into  others  whose  co- 
efficients shall  be  whole  numbers,  that  of  .r"  being  unity : 

6.  x^^%x-l  =  ^. 

8.   ^«  _  3  .,;2  +  ^  ^  _  2  ^  0. 

9.      X^  —    \  X^  —    I  ^2   +    O.I  JC   +    y^ly^   =   0. 


28o  ALGEBRA. 

392.  A  Commensurable  real  root  is  one  that  can  be 
exactly  expressed  as  a  whole  number  or  a  rational 
fraction. 

An  Incommensurable  real  root  is  one  whose  exact 
expression  involves  surds. 

Thus,  of  the  equation  (r  —  5)  (^  —  ^)  (;r  —  yy/2)  (x  +  y^i)  =  0, 
5  and  ^  are  commensurable  roots,  and  y/2  and  —  y/2  imom- 
mensurable. 

393.  Integral  Roots.     If  the  coefficients  ^  F  (x)  are 

all  wJiole   numbers^   any  commensurable    real  rcot  of 

F(x)  =  0  is  a  whole  number  and  an  exact  division  of  p,,. 
s 
Suppose-,  a  rational  fraction  in  its  lowest  terms,  to 

be   a   root  of  F{x)  =  ^\    then,  by   substitution,  we 
have 


-n  +A  ^33  +A^-;73T,  +  •••  +A  =  0.  (.) 


Multiplying  by  /"    ',  and  transposing,  we  have 

^'  =  -(A«f""'+A^^""'  +  ---+A/""')-      (2) 

Now  (2)  is  impossible,  for  its  first  member  is  a 
fraction  in  its  lowest  terms,  and  its  second  member 
is  a  whole  number. 

Hence,  as  a  rational  fraction  cannot  be  a  root,  any 
commensurable  root  must  be  a  w^hole  number. 

Next,  let  a  be  an  integral  root  oi  F  (x)  =  0. 

Substituting  a  for  x,  transposing  /„,  and  dividing 
by  a,  we  have 


.      THEORY   OF  EQUATIONS.  28 1 

The  first  member  of  (3)  is  "integral;  hence  the 
quotient/,,  4-  «  is  a  whole  number. 

Thus,  any  commensurable  root  of  x-^  —  6  x^  -\-  10  jr  —  8  =  0 
must  be  ±1,  ±2,  ±  4,  or  ±  8  ;  for  these  are  the  only  exact 
divisors  of  —  8. 


394.  The  Limits  of  the  Roots  of  an  equation  are 
any  two  numbers  between  which  the  roots  lie. 

The  limits  of  the  real  roots  may  be  found  as  follows: 

Superior  Limit.  In  evaluating  F{fi)  in  Example  i 
of  §  381,  the  sums  are  all  positive,  and  they  evi~ 
dently  would  all  be  greater  for  x  >  6.  Hence  F{x) 
can  vanish  only  for  x  <  6\  and  therefore  all  the 
real  roots  of  F{x)  —  ^  are  less  than  6. 

Hence,  if  in  computing  the  value  of  Y  (c),  c  being 
positive,  all  the  sums  are  positive,  the  real  roots  of 
F  (x)  =  0  are  all  less  than  c. 

Inferior  Limit.  In  evaluating  F{—6)  in  Example 
2  of  §  38r,  the  sums  are  alternately  —  and  +,  and 
they  evidently  would  all  be  greater  numerically  for 
X  <  —  6.  Therefore  all  the  real  roots  of  F  {x)  =  {) 
are  greater  than  —  6. 

Hence,  if  in  computijig  the  value  ^F(b),  b  being 
negative,  the  sums  are  alternately  —  and  +,  all  the  real 
roots'  of  ¥  (x)  =  0  are  greater  than  b. 

Therefore,  if  its  coefficients  are  alternately  +  and 
— ,  F  {x^  =  0  cannot  have  any  negative  roots. 


282  ALGEBRA. 

Example.     Solve  x*  *+  2  x^  —  i^  x^  —  14  x  +  24  =  0. 

In  evaluating  /^(4),  the  sums  are  all  +  ;  and  in  evaluating 
7^{—  5),  the  sums  are  alternately  —  and  +  ;  hence  the  real  roots 
of  I^{x)  =  0  lie  between  —  5  and  4. 

By  §  393,  the  commensurable  roots  are  integral  factors  of  24. 
Hence  any  commensurable  root  must  be  ±  i,  +  2,  ±  3,  or  —  4. 

The  work  of  determining  which  of  these  numbers  are  roots 
may  be  arranged  as  below : 

I        +2        -  13        -  14        +  24  l_r^ 
+  1         4-3         -  10        -  24 


I  -}-  3  -  10  -  24    |-2 

—  2—2+24 
+  1  -  12  0 

Hence  J^(x)  is  divisible  by  x  —  i,  the  quotient  x^  +  ^  x^ 

—  10  jr  —  24  is  divisible  by  :r  +  2,  and  the  depressed  equation  is 

X^+  X-  12  =  0, 

of  which  the  roots  are  evidently  3  and  —4. 

Therefore  the  required  roots  are  i,  —  2,  3,  —4. 

EXERCISE  55. 

1.  Show  that  the  real  roots  of  :v:^  —  2  .:v  —  50  =  0  lie 
between  —  2  and  4. 

2.  Show  that  any  commensurable  real  root  of  x*  —  ^x^ 

—  1S^  ~  loooo  =  0  is  ±  I,  ±  2,  ±  4,  ±  5,  ±  8,  or  10. 

3.  Show  that  the  real  roots  of  :v^  +  2  ^*  +  3  :t^  +  4  x^ 
+  5  jc  —  54321  =  0  lie  between  —  2  and  9,  and  that  any 
commensurable  real  root  must  be  ±  i  or  3. 

4.  Any  commensurable  root  ofx^ —  15^^^+  io;t:+  24  =  0 
must  be  one  of  what  numbers  ? 

5.  Find  the  roots  of  the  equation  in  Example  4. 


THEORY   OF   EQUATIONS.  283 

Solve  each  of  the  following  equations,  and  verify  the  roots 
of  each  by  §  387 : 

6.  x^  —  4  X*  —  16  .T^  4-  1 12  jc^  —  208  X  +  128  —  0. 

7.  ^*  —  4  a:^  —  8  ^  +  32  =:  0. 

8.  x""  —  zx^  ^  X  ^  2^^, 

9.  x^  —  d  x"^  -\-  \\  X  —  6  =  0. 

10.  jc''  —  9  ;c^  +  1 7  x'^  4-  27  Ji  —  60  =  0. 

11.  x^  —  d  x"^  -\-  \o  X  —  8  =  0. 

12.  x"^  —  6  x^  +  24^:1;  —  16  =  0. 

13.  ^^  —  3  x^  —  9  jt-^  +  2 1  .r^  —  10  jc  +  24  =  0. 

14.  x"^  —  x^  —  39  x"^  +  24  a:  +  180  =  0. 

15.  ^»  +  5  -^"^  —  9  -^  —  45  =  0- 

16.  ^*  —  3  jc*  —  14  Ji:^  +  48  .r  —  32  =  0. 

17.  x^  -{-  x^  —   14^^  —  14  a*  +  49  .V*  +  49  ^'^  — 36  a: 
-  36  =  0. 

18.  ^«+5  ^^— 81  a;*— 85  JtH  964^2^  780 ;t:— 1584  =  0. 

19.  ^'  —  8  a:^  +  13^  —  6  =  0. 

20.  x^  -\-  2  x^  —  23  jc  —  60  =  0. 

21.  ^^  —  45  :r^  —  40  j«r  4-  84  =  0. 

22.  x^'-'jx'  +  iix'-  Tx'^  i4x''-28x-\-  40  =  0. 


284  ALGEBRA. 

Solve  the  following  equations  by  first  transforming  them 
into  others  whose  commensurable  roots  are  whole  numbers : 

23-   ^^  —  t  ^''  —  ih  ^  +  /ff  =  0- 
24.   8  Jt^  —  2  ..-  —  4  ^ '+  1=0. 

26.  gx^  —  (^x^i-^x^  —  ^x+^  =  0, 

27.  S  x^  —  26  o;"^  +  1 1  -^  +  ro  =  0. 

28.  X*  —  6  x^  +  g^  x^  —  ;^  X  -{-  47}  =  0. 

395.  Equal  Roots.  Suppose  the  equation  F(x)  =  0 
has  r  roots  equal  to  a,  and  let 

J^(x)  =  {x  —  aycfi{x);  (i) 

then  F'  (x)  =  r  {x  -  (i)'-'' cIj(x)  +  (x  —  aY  cfj' (x) .     (2) 

From  (i)  and  (2)  it  is  evident  that  {x  —  ay~'  is 
a  common  factor  of  i^(;r)  and  F'  (;r). 

Hence  if  F (;l)  =  0  has  r  roots  equal  to  a,  (x  —  ay~  ^ 
will  be  a  factor  of  the  H.C.D.  of  F (^x)  and  F' {x). 
Any  linear  factor  will  occur  once  more  in  F  (x)  than 
in  the  H.C.D.  o{F{x)  and  F'  (x). 

Example  i .    Solve  x*  —  1 1  jr^  +  44  ;i-2  —  76  ^r  +  48  =  0  (i) 
Here  F (x)  ~  x^  -  i\  x^  -^  44 x"^  -  76 x  +  4S  -, 

.-.     F'(x)  =  4  ;r3  -  33  x^  +S8x-  76. 

By  the  method  of  §94  we  find  the  H.C.D.  of  F(x)  and 
F'{x)  to  be  ,r  —  2  ;  hence  two  roots  of  (i)  are  2  each. 

%  §  387,  the  sum  of  the  other  two  roots  is  7,  and  their  pro- 
duct 12  ;  hence  the  other  two  roots  are  4  and  3. 


THEORY   OF   EQUATIONS.  285 

Example  2.   Solve 

x'  +  S^  +  dx^^dx^-  i5;i-3-3;i'2-|-  8:r+.4  =  0.      (i) 

Here  the  H.  C.  D.  of  F{^x)  and  F{x)  is 

x^  -\-  -i^x^  ^-  x'^  —  3  -r  —  2.  (2) 

The  H.  C.  D.  of  function  (2)  and  //s  derivative  is  x  +  i  ; 
hence  (x  +  i)-^  is  a  factor  oi  (2).     liy  factoring  we  obtain 

x^  +  Zx^  +  x'2-3x-2  ^{x-^  1)2  (x  -f-  2)  (^  -  I).      (3) 

Hence  three  roots  of  (i)  are  —  i   each,  two  —  2  each,  and 
two  I  each. 

EXERCISE  56. 

Solve  the  following  equations,  each  having  equal  roots : 

1.  X*—  i4x^  +  61  x^  —  84 ^  +  36  =  0. 

2.  x^  —  'J  x"^  ■\-  16  X  —  12  =  0. 

3.  a:*  —  24  ^^  +  64  ;t:  —  48  =  0. 

4.  AT*  —  1 1  a:^  +  i8  a:  —  8  =  0. 

5.  X*  +  13  ;f»  +  Z2>  ^^  +  31  -^  +  10  =  0. 

6.  x^  —  2  x^  +  3  ^^8  —  7  .t2  +  8  x  —  3  =  0. 

7.  ;c*  —  12  :r^  +  50  A-2  _  84  .r  +  49  =  0. 

8.  A*  +  3  a^  -  6  :^*  -  6  :i8  +  9  ^'  +  3  ^  -  4  =  0. 

9.  Show  that  the  equation  ^^  +  3  ffx  +(9  =  0  will  have 
two  equal  roots,  when  4  H^  +  (9^  =  0. 

10.  If  4  r  =/2^  find  the  roots  of  x*  — /  jc'^  +  r  =  0. 


286  ALGEBRA. 

396.  If  only  two  of  the  roots  of  a  higher  numeri- 
cal equation  are  incommensurable  or  imaginary, 
the  commensurable  real  roots  may  be  found  by  the 
methods  already  given,  and  the  equation  depressed 
to  a  quadratic,  from  which  the  other  two  roots  are 
readily  obtained. 

When  a  higher  numerical  equation  contains  no 
commensurable  real  root,  or  when  the  depressed 
equation  is  above  the  second  degree,  the  following 
principle  is  useful  in  determining  the  number  and 
situation  of  the  real  roots. 

397.  Change  of  Sign  of  F(x).  7/"  F(b)  and  F(c) 
have  unlike  signs,  an  odd  number  of  roots  o/F  (x)  —  0 
lies  between  b  and  c. 

If  X  changes  continuously,  then  F{x)  will  pass 
from  one  value  to  another  by  passing  through  all  in- 
termediate values  (§  255).  Therefore  to  change  its 
sign,  F{x)  must  pass  through  zero ;  *  for  zero  lies 
between  any  two  numbers  of  opposite  signs. 

Hence  \{  F{b)  and  F{c)  have  opposite  signs,  F{x) 
must  vanish,  or  equal  0,  for  one  value,  or  an  odd 
number  of  values,  o{ x  between  b  and  c. 

If  F{b)  and  F {c)  have  like  signs,  then  we  know 
simply  that  either  no  root,  or  an  even  number  of 
roots,  of  F  (x)  =  0  lies  between  b  and  c. 

*  A  function  may  change  its  sign  by  passing  through  infinity 
(§  254)  ;  but  evidently  F{x)  or  any  other  integral  function  of  x  can- 
not become  infinite  for  a  finite  value  of  x. 


THEORY   OF   EQUATIONS.  28/ 

Example.    Find  the  situation  of  the  real  roots  of 

By  §  394  we  find  that  all  the  real  roots  lie  between  —  2  and  6. 

Herei^(-2)=-4,         /^(O)  =  +  8,         7^(4)  =  - 16, 

/r(_i)  =  +  9,        yr(,)  =  _i,        /^(5)  =  +  3- 

'  Since  F(—  2)  and  F(~  i)  have  unlike  signs,  at  least  one 
root  of  F(x)  =  0  lies  between  -  2  and  -  i.  For  like  reason  a 
second  root  lies  between  0  and  i,  and  a  third  between  4  and  5. 
Hence  the  roots  are  -  (i-  +),  0-  +,  and  4.  +. 

398.  Every  equation  of  an  odd  degree  has  at  least 
one  real  root  whose  sign  is  opposite  to  that  of  the  knoivn 
term  p„. 

\iF(x)  is  of  an  odd  degree,  then 

ir(_  x)  =  -  X,  /^(O)  =/,„  7r(  +  ^)  ^  +  ^^. 

Hence,  if /„  is  positive,  one  root  of  F{x)  —  0  lies 
between  0  and  —  x  (§  397) ;  and  if  /„  is  negative, 
one  root  lies  between  0  and  +  x. 

399.  Every  equation  of  an  even  degree  in  which  p„ 
is  negative  has  at  least  one  positive  and  one  Jtegative 
real  root. 

Here  F{-^)=:  +  x,  7^(0)  is  — ,  7^(+ x)  rn  +  x. 

Hence  one  root  of  FCx)  =  0  lies  between  0  and 
—  yo,  and  another  between  0  and  +  x. 


288  ALGEBRA. 

EXERCISE  57. 

Find  the  first  figure  of  each  real  root  of 

1.  x^  +  x^—2x—i=0.         6.    x^—4x^  -  6x  =  ~S, 

2.  .:r^— 3 J\;^— 4:^+11  =0.         7.    ^'*— 4^^  —  3 Jt:  =  —  27. 

3.  :v^— 4^'— 3  x+ 23  =  0.        8.    ^^  +  :r  —  500  =  0. 

A.    X^  —  2X  —  ^  =  0.  9.    ::c^+ 10  a:^+ 5^  =  260. 

5.    20;(;^—  24;^;^+  3  =  0.      10.    :t:^+  3  ^^  +  5  ^  =  178. 
II.    :v^  —  11727^  +  40385  =  0. 

Sturm's  Theorem. 

*400.  The  object  of  Sturm's  theorem  is  to  deter- 
mine the  number  and  situation  of  the  real  roots  of 
any  numerical  equation.  Though  perfect  in  theory, 
Sturm's  theorem  is  laborious  in  its  application. 
Hence,  when  possible,  the  situation  of  roots  is  more 
usually  determined  by  the  method  of  §   397. 

*401.  Sturm's  Functions.  Let  F  {x)  =  0  be  any 
equation  from  which  the  equal  roots  have  been  re- 
moved, and  let  F'  (x)  denote  the  first  derivative  of 
F{x).  Treat  F{x)  and  F  {x^  as  in  finding  their 
H.  C.  D.,  with  this  modification,  that  the  sign  of  each 
remainder  be  changed  before  it  is  used  as  a  divisor, 


THEORY   OF   EQUATIONS.  289 

and  that  no  other  change  of  sign  be  allowed.  Con- 
tinue the  operation  until  a  remainder  is  obtained 
which  does  not  contain  x,  and  change  the  sign  of 
that  also.  Let  F^  {x),  h\  {x),  . . .,  F^  (,r^)  denote  the 
several  remainders  with  their  signs  changed;  then 
F{x^,  P{x\  F,{x),  F^{x),  ...,  /s,,  Ot^)are  called 
St  linn  s  Functions. 

F(x)  is  the  primitive  function,  and  P  {x)y  F^{x)y 
...,  F,„  {x^^)  are  the  auxiliary  functions.  We  use  ^  in- 
stead of  Ji'  in  F„^  {^)y  since  F,^  (,r^)  does  not  contain  x. 

Example.  Given  x^  —  ;^x^  —  4x -\-  13  =  0;  find  Sturm's 
functions. 

Here  F (x)  ^  x^  -  3  x^  -  4  x  +  13  ; 

.-.    P(x)  =  3x'^-6x-4. 

Dividing;  F(x)  by  P(x),  first  multiplying  the  former  by  3 
to  avoid  fractions,  we  find  that  the  first  remainder  of  a  lower 
degree  than  the  divisor  is  —  I4;r  +  35.  Changing  the  sign  of 
this  remainder  and  rejecting  the  positive  factor  7,  we  have 

F,(x)^2x-5. 

Proceeding  in  like  manner  with  3  or^  —  6ar  —  4  and  2  x  —  ^, 
we  find  the  next  remainder  to  be  —  i ;  hence  F^^x^)  :i:  +  i. 

If  an  equation  has  equal  roots,  the  process  of  find- 
ing Sturm's  functions  will  discover  them,  and  then 
we  can  proceed  with  the  depressed  equation. 

*402.  A  Variation  of  sign  is  said  to  occur  when  two 
successive  terms  of  a  series  have  unlike  signs ;  and  a 
Permanence,  when  they  have  like  signs. 


290  ALGEBRA. 

Thus,  if  the  signs  of  a  series  of  quantities  are  +  -I h  + 

— t- ,  there  are  four  variations  and  three  permanences  of  sign. 
Again,  in  the  Example  of  §  401,  we  have 

J^  (x)  =  x^  -  3 x^  -  4x  +  IS,  Fi(x)  =  2x-  5, 

F\x)  =  3x^~6x-4,  F^{x^)  =  +  i. 

When            F{x)      F' {x)      F^{a')  F^{x^) 

X  =  0        +             —              —  +2  variations. 

x=z  3        -\-            +              +  +0  variations. 

*  403.  Sturm's  Theorem.  7/"  F  (x)  =  0  has  no  equal 
roots  and  b  be  substituted  for  yi  in  Sturm  s  functions, 
and  the  number  of  variations  noted,  and  then  a  greater 
number  c  be  substituted  for  x  and  the  7iumber  of  varia- 
tions noted ;  the  first  number  of  variations  less  the 
second  equals  fhe  number  of  real  roots  of  F  (x)  =  0 
tJiat  lie  between  b  and  c. 

(i.)  Since  each  of  Sturm's  functions  is  an  integral 
function,  to  change  its  sign  a  Sturmian 
function  must  vanish  (§  255). 

(ii.)  Two  consecutive  functions  cannot  vanish  for 
the  same  value  of  x. 

For  if  Fi  {x)  and  Fz  (x)  both  vanished  when  x  —  a, 
each  would  contain  the  factor  x  —  a.  Hence,  by  §§94 
and  401,  F{x)  and  F'  {x)  would  have  the  common 
factor  X  —  a ;  whence,  by  §  395,  F{x)  =  0  would  have 
equal  roots,  which  is  contrary  to  the  hypothesis. 

(iii.)   When    any   auxiliary   function   vanishes,   the 
two  adjacent  functions  have  opposite  signs. 


THEORY   OF   EQUATIONS.  29I 

Let  the  several  quotients  obtained  in  the  process 
of  finding  Sturm's  functions  be  represented  by  q^y  q^^ 
^3,  ...;   then  by  principles  of  division  we  have 

F'{x)=F,{x)q,-F,{x), 
F,{x)=F,(x)q,-F,{x\ 

Hence,  if  any  auxiliary  function,  as  R,  (;r),  vanishes 
when  X  —  a,  from  the  third  identity  we  have 
F,{a)=-F,{a). 
(iv.)  The  number  of  variations  of  sign  of  Sturm's 
functions   is   not   affected   by  a   change   of 
sign  of  any  of  the  auxiliary  functions. 

Suppose  ^^2  C-^)  to  change  its  sign  when  x  —  a  ; 
then,  by  (i.)  and  (ii.),  neither  Fx{x)  nor  F^^x)  can 
change  its  sign  when  x  =  a.  Hence  F^  {x)  and  Fz  {x) 
will  have  the  same  signs  immediately  2Xx.^x  x  —  a  that 
they  had  immediately  before,  and  by  (iii.)  these 
signs  will  be  unlike. 

Now,  whichever  sign  be  put  between  two  unlike 
signs,  there  is  one  and  only  one  variation.  Hence  the 
change  of  sign  of  F,^  {x)  does  not  affect  the  number 
of  variations  of  sign.  The  same  holds  true  of  any 
other  auxiliary  function  except  F„,  {x^),  which,  being 
constant,  cannot  change  its  sign. 

(v.)  If  X  increases,  there  is  a  loss  of  one,  and  only 
one,  variation  of  sign  of  Sturm's  functions 
when  F  {x)  vanishes. 


292  ALGEBRA. 

When  F(x)  vanishes,  F\x)  is  +  or  — . 

li  F'  (x)  is  +,  F{x')  by  §  238  is  increasing  when  it 
vanishes,  and  therefore  must  change  its  sign  from  — 
to  +.    Hence,  immediately  before  FQv)  vanishes,  we 

have  the  variation 1-,  and  immediately  afterward 

the  permanence  +  +. 

\{  F'  (x)  is  — ,  F(x)  is  decreasing  when  it  vanishes, 
and  therefore  must  change  its  sign  from  +  to  •— . 
Hence,  when  Fi^x')  vanishes,  the  variation  +  —  be- 
comes the  permanence  —  — . 

Whence  there  is  a  loss  of  one  variation  of  Sturm's 
functions  when  F(x)  vanishes,  and  only  then. 

Therefore  the  number  of  variations  lost  while  x 
increases  from  b  to  c  \s  equal  to  the  number  of  roots 
of  F(x^  =  0  that  lie  between  b  and  c. 

Example  i.    Determine  the  number  and  situation  of  the 
real  roots  of  the  equation  x^  —  2>  x'^  —  ^x  -\-  13  =  0. 
By  §  394,  all  the  real  roots  lie  between  —  3  and  4. 

Here  F{x')  :=3  jir^  —  3  jr^  —  4  ;ir  +  13,  F^  (.r)  =  2  ;r  —  5, 

F'  {x)  Ez  3  ^2  _  6^-  -4,  F^{x^)  ^  +  I. 

Beginning  at  ;ir  =  —  3,  we  find  the  following  table  of  results  : 

F,{x)      F,{x) 

—  +3  variations. 

—  +2  variations. 

—  +2  variations. 
+              +0  variations. 

Hence  there  is  one  negative  root  between  —  3  and  —  2  (§  403), 
and  two  positive  roots  between  2  and  3.  To  separate  the  two 
positive  roots,  we  substitute  in  the   Sturmian  functions  some 


When             F{x) 

F\x) 

x=-z       - 

+ 

X=~2         + 

+ 

:r=      2       + 

- 

x=      3       + 

+ 

THEORY    OF   EQUATIONS.  293 

value  of  X  between  2  and  3,  as  2.5.     When  x=  2.5,  the  suc- 
cession of  signs  is 0  +,  which  gives  but  one  variation, 

whether  0  has  the  sign  +  or  — ;  hence  one  positive  root  lies 
betweeti  2  and  2.5,  and  the  other  between  2.5  and  3. 

Example  2.  Find  the  number  and  situation  of  the  real 
roots  of  2x^  —  13^-2+  10  X  ~  19  =  0. 

Sturm's  theorem  may  be  applied  to  an  equation  in  this  form, 
since  there  is  nothing  in  its  demonstration  that  requires  the 
coefficient  of  x"  to  be  unity. 

By  §  394,  the  real  roots  lie  between  —  4  and  -f  3. 

Here  F{x)^2x*  —  I'^x^  +  \ox  ~  i<), 

F'{x)^2(/[x^-iZx+s\ 
F^ix)^i2,x''-iSx  +  z^' 

Since,  by  §  148,  the  roots  of  F^{x)  ~  13  ;i'2  -  15  jt-  +  38  =  0 
are  imaginary,  F-^{x)  cannot  change  its  sign  for  any  real  value 
of  xj  hence  there  can  be  no  loss  of  variations  beyond  F^  (:r), 
and  it  is  unnecessary  to  obtain  F^{x)  and  F^{x'^). 

When 


in 

F(,x) 

F'{x) 

-^iW 

:r=-4 

+ 

- 

+ 

2  variations. 

x=-^ 

+ 

- 

+ 

2  variations. 

X  =  —.2. 

— 

- 

+ 

I  variation. 

X=        2 

- 

+ 

+ 

I  variation. 

x=     3 

+ 

+ 

+ 

0  variations. 

Hence  there  are  two  real  roots,  one  of  which  is  —  (2.  +)  and 
the  other  2.  +. 

Example  3.    Find  the  number  of  the  real  roots  of  the  cubic 
x^  +  2>^^-+G  =  0.  (I) 

Here  F{x)  ^  x^  ^- ^  Hx  ^  G,  F^  {x)  ~~2  Hx  -  G, 
F'  (A)  :    3  {x^  +  H),  F.,  {x)  =  -  (6^2  +  4  ^3). 


294  ALGEBRA. 

If  G^  +  4l/^>0,  H  may  be  either  +  or  — ;  so  that 
When  F{x)      F' {x)      F^{x)      F^{x) 

X  =  —  T>       —  +  ±  —       2  variations. 

:f  =  +jo4-  +  T  —       I  variation. 

Hence  when  6^^  +  4  H^  \s  positive,  only  ojte  root  is  real. 
If  6^-^  +  4  //3  <  0,  evidently  //  is  —  ;  so  that  we  have 

.r  =  —  30—  +  —  +3  variations. 

;jr=:+x+  +  +  +       0  variations. 

Hence  when  G^  +  4//^  is  negative,  all  ///r<?^  roots  are  real. 

If  G^2  ^4iY3  =  0,  /^(;r)and  7'^'(;r)  have  2Hx+  G^  as  a  C.  D.; 
hence,  by  §  395,  two  roots  are  —  G  -^  1  H  each.  By  §  387  the 
third  root  \s  G  -^  H. 

EXERCISE  58. 

Find  the  first  figure  of  each  real  root  of 

1.  x^  -\-  2  x^  —  T,  X  —  ().  3.   x^  —  ^  x'^  ■\-  ^x  =  1. 

2.  .^^  —  2  a:  —  5  =  0.  4.    x^  ~  x"^  —  2  X  =  —  1. 

5.  jc*  —  4  .;c^  —  3  .r  +  23  =  0. 

6.  x^  -{-  '^  x^  —  4  x'^  —  -i  \  X  -\-  ^  —  0. 

7.  jf'*  —  2  jc^  —  5  ^^  +  10  jc  —  3  =  0. 

8.  jc^—  lo.^^  +  6.^  +  I  =0. 

9.  Show  that  in  general  there  are  «  4-  i  Sturmian  func- 
tions. 

10.  Show  that  all  the  roots  of  F{x)  =  0  are  real  when 
the  first  term  of  each  of  the  n  +  1  Sturmian  functions  has 
a  positive  coefficient. 


theory  of  equations.  295 

Transformation  of  Equations. 

404.  In  solving  an  equation  it  is  often  advantageous 
to  transform  it  into  another  whose  roots  shall  have 
some  known  relation  to  those  of  the  given  equation. 
For  one  case  of  transformation  see  §  391. 

405.  To  transform  aii  equatioji  into  anotJier  whose 
roots  shall  be  those  of  the  first  with  their  signs 
changed. 

If  in  F(x)  =  0,  we  put  x  =  —  x^,  we  obtain 

F{-x,)  =  (-x,)"  +/,(-^J"-i  +  ... 

+  A-i(-^\)+A  =  0.      (i) 

Since  x  =  —x^,  the  roots  of  F{x)  =  0  and  F{—Xj)  =  0 
are  numerically  equal  with  opposite  signs. 

If  in  (i)  we  perform  the  indicated  operations,  the 
terms  will  be  alternately  +  and  —  or  —  and  +. 

Hence,  to  effect  the  required  transformation,  change 
the  signs  of  all  the  terms  co7itaining  the  odd  powers  of 
X,  or  of  those  containing  the  even  powers. 

Thus,  the  roots  of  :i^  -  ;i-2  -f-  3  r  +  6  =  0  are  numerically 
equal  to  the  roots  of  x*  —  x^  —  ^x-  +  6  =  0,  but  opposite  in 
sign.  The  same  is  true  of  the  roots  of  x^  —  y  x*  +  x^  +  i  =  0 
and  x^+  7x^  +  x^-  I  =0. 

406.  To  transform  an  equation  into  another  whose 
roots  shall  be  the  reciprocals  of  those  of  the  first. 


296  ALGEBRA. 

If  in  F{x)  —  0  we  put  x  —  \  ^  x^,^^  obtain 


which  is  evidently  the  equation  required. 

Multiplying  (i)  hy  x{\  and  reversing  the  order  of 
the  terms,  we  obtain 

+  A^i'+A-^i  + ^  =  0.    (2) 

Hence,  to  effect  the  required  transformation,  write 
the  coefficients  in  the  reverse  order. 

Thus,  the  roots  of  2  ,r3  -  3  ;r2  —  4  r  +  $  =  0  are  the  recip- 
rocals of  the  roots  of  5  ^-^  —  4  Jt-^  —  3  :ir  -f  2  =  0. 

407.  Infinite  Roots.  If /„  —  0,  one  root  of  F{x)  —  0 
is  0,  and  therefore,  by  §  406,  the  corresponding  root 
of  F(^\  -j- jiTi )  =  0  is  I  -i-  0,  or  infinity. 

That  is,  if  hi  an  equation  the  coefficient  of  x"  is  0, 
one  root  is  infinity. 

Similarly,  if  the  coefficients  of  x^  and  x"~^  are  both  0, 
two  roots  are  infinity  ;  and  so  on. 

Thus    in   the   linear   equation   a  x  —  b^    if  «:  =  0,   the   root* 
b  ^  a—  y:>\   if  ^  =  0,  the  root  b  ^  a  —  b  ^  ^,  or  infinity. 
Again,  the  roots,  in  §  145,  of  the  quadratic  equation 

ax'^-\-bx^-c  =  ^, 

by  rationalizing  the  numerators,  may  be  put  in  the  forms, 

ic  ic 


b-^b^-^ac  -b+  ^b'^  -4ac 


THEORY  OF  EQUATIONS.  297 

Now  if  ^  =  0,  then  a-:^  —  c-^b  and  ^  =  >d  ;  if  «  =  0,  one  root 
is  finite,  and  the  other  is  infinity. 

If  ^  =  0  and  ^  =  0  also,  then  a  =  do  and  /3  =  x  ;  if  «  =  ^  =  0, 
both  roots  are  infiiiity. 

408.  To  transform  an  equation  into  another  whose 
roots  shall  be  those  of  the  first  diminisJied  by  a  given 
quantity. 

If  in  the  equation 

Fix)  =  X"  +A^"~'  +  A^"~'  +  ••• 

+  A-i^-+A  =  0,  (i) 

we  put  X  —  x^  ■\-  h,  we  obtain 

F{x,  -\-h)  =  {x,  +  hy  +  A  ('^i  +  ^)"-'  +  ... 

+  A-i(-^i  +  /0+A-0.  (2) 

As  X  =  Xy  -\-  hy  or  x^=x  —  //,  the  roots  of  (2)  equal 
those  of  (i)  diminished  by  h,  h  being  either  positive 
or  negative. 

Hence,  to  eflfect  the  required  transformation,  sub- 
stitute X,  +  h  for  X,  expandy  and  reduce  to  the  form  of 
F  (x)  =  0. 

409.  Computation  of  the  Coefficients  of  F(jc^  +  //). 
Since  F{x^  +  //)  may  be  reduced  to  the  form  of 
F{x),  put 

F{x^  +  h)  =  Xj"  +  qiXi"-^  +  q^x{'-'^+  ...  +  $^„-i^i  +  q„. 
Substituting  x  —  h  for  Xi,  we  obtain 

F{x)  =  (x-h)"  +  q,{x-/iy'-^  -i-  ...-{-  q„_,(x--/i)  +  q^. 


298  ALGEBRA. 

Dividing  F  {x)  by  x  —  //,  we  obtain 

(^^  _  /,)«-!  +  g,{x-  hy-^  +  ..•  +  qn.-i  (i) 

as  the  quotient,  and  q,,  as  the  remainder. 

Dividing  the  quotient  (i)  hy  x  —  h,  we  obtain 

as  the  quotient,  and  q„_^  as  the  remainder. 

The  next  remainder  will  be  qn-i\  the  next  qn-z\ 
and  so  on  to  q^. 

The  last  quotient  will  be  the  coefficient  of  ;tr". 

Hence,  if  F  (x)  be  divided  successively  by  x  —  h, 
the  successive  remainders  and  the  last  quotient  will  be 
the  coefficients  o/F  (x^  +  h)  in  reverse  order. 

Example.  Transform  the  equation  x^  —  ^x^  —  2x+s  =  ^ 
into  another  whose  roots  shall  be  less  by  3. 

The  work  of  dividing  /^(x)  successively  by  ;ir  —  3  to  compute 
the  coefficients  of  i^(^i  +  3)  may  be  arranged  as  below : 


I 

-3 

—  2 

+  5     |3 

+  3 

+  0 

-6 

I 

0 

-  2 

-^  =  q 

+  3 

+  9 

--^2 

I 

+  3 

+  7  = 

+  3 

I  +  6  =  ^1 

Hence  the  transformed  equation  is 

F(xi  +  3)      Xj^  +  6xi^  + 7x^-1=  0. 


THEORY  OF  EQUATIONS.  299 

410.  Equation  Lacking  any  Term.  If  the  binomials 
in  equation  (2)  of  §  408  be  expanded,  the  coefficient 
of  or/'"'  will  evidently  be  ;///  +A;  hence  if  we  put 
n  h  -\-  py  =  0,  or  //  =  — /i  -^  ;/,  the  transformed  equa- 
tion will  lack  the  term  in  x"~\ 

In  like  manner  an  equation  can  be  transformed 
into  another  which  shall  lack  any  specified  term. 

Example.  Transform  the  equation  x^  — 6x'^-\-^x-\-$  —  0 
into  another  lacking  the  term  in  xK 

Here  /^  =  —  6,  ;/  =  3  ; 

.'.    h  =  —  p^  -^  n  =  2.. 

Transforming  the  given  equation  into  another  of  which  the 
roots  are  less  by  2,  and  writing  x  for  x^,  we  obtain 

:i-8-8jr-3  =  0, 

an  equation  which  lacks  the  term  in  x-.    ■ 

EXERCISE   59. 

Transform  each  of  the  following  equations  into  another 
having  the  same  roots  with  opposite  signs : 

1.  x^  +  a:^  —  a-2  —  5  a:  +  7  =  0. 

2.  ;c^  -  7  a:^  —  5  ;t:2  +  8  =  0. 

3.  :j;^  —  6  ^*  —  7  ^^^  +  5  ^  =  3. 

/[.  x'  —  T  X*  ■\-  ^x"^  —  ^  X  -{-  2  =  0, 


300  ALGEBRA. 

Find  the  equation  whose  roots  are  the  reciprocals  of  those 
of  each  of  the  following  equations  : 

5.  x^  —  'jx^  —  4x+2=0.     7.   x^  —  x^  +  ^x'^+  S  =  0. 

6.  x^—8x^  +  T  =  0.  8.   x^  —  x^  +'jx^  +  g  =  0. 

Transform  each  of  the  following  equations  into  another 
whose  roots  shall  be  less  by  the  number  placed  opposite 
the  equation : 

9.   ^^  —  3  ^^  —  6  =  0.  5.  . 

10.   x^  —  2  x'^  -\-  ^  x^  -{-  4.  X  —  y  =  0.  4. 

11.:^*  —  2  JC^  +  3  Jt:^  +  5  -^  +  7  =  0.  —  2. 

12.    X*  —  18  ::t:^  —  ^2  x^  -\-  I'j  X  -\-  ig  =  0.  5. 

13-    5  ^*  +  28  ^^  +  51  j\;^  +  32  jc  —  I  =  0.  —  2. 

Transform  each  of  the  following  equations  into  another 
which  shall  lack  the  term  in  x''^ : 

14.  x^  —  ^  x^  +  S  ^  +  4  =  ^' 

15.  x^  —  6x''+  8x—  2  =  0. 

16.  x^  +  6  x^  —  J  X  —  2  =  0, 

I  7.    j:!::^  —  9  ^^  +  12  a:  +  19  —  0. 

411.  Horner's  Method  of  Solving  Numerical  Equa- 
tions. By  this  method  any  real  root  is  obtained, 
after  Its  situation  has   been  determined.     The   main 


THEORY   OF  EQUATIONS.  3OI 

principle  involved  is  the  successive  diminution  of 
the  roots  of  the  given  equation  by  known  quantities, 
as  explained  in  §  408. 

Thus,  suppose  that  one  root  of  /^(x)  =:  0  is  found  to  lie 
between  40  and  50;  to  find  this  root  we  transform  the  equation 
yr  (i-)  =:  0  into  another  whose  roots  shall  be  less  by  40,  and 

"•^'^i"  ^(4o  +  r,)  =  0,  (i) 

of  which  the  positive  root  sought  is  less  than  10. 

If  this  root  is  found  to  lie  between  6  and  7,  we  transform 
equation  (i)  into  another  whose  roots  shall  be  less  by  6,  and 
obtain  ^  j-^^  ^  (g  _^  ^^>j-|  =  /r  (46  +  :,,)  =  0,  (2) 

of  which  the  positive  root  sought  is  less  than  i. 

If  this  root  lies  between  05  and  0.6,  we  transform  equation 
(2)  into  another  whose  roots  shall  be  less  by  o  5,  and  obtain 

7^(46.5 +  ;ro)  =  0,  (3) 

of  which  the  positive  root  sought  is  less  than  o.i. 

First,  suppose  this  root  of  (3)  to  be  o  03,  we  then  have 
-^(46-53)   -  0 ;  that  is,  one  root  of  F(x)  =  0  is  46.53. 

Next,  suppose  this  root  of  (3)  to  lie  between  0.03  and  o  04  ; 
then  it  follows  that  one  root  of  F{.x)  =  0  lies  between  46.53 
and  46  54. 

By  transforming  equation  (3)  into  another  whose  roots  shall 
be  less  by  0.03,  the  thousandths  figure  of  the  root  can  be  found  ; 
and  so  on. 

By  repeating  these  transformations  we  can  evidently  obtain 
a  root  exactly,  or  may  approximate  to  any  root  as  nearly  as  we 
please. 

412.  One  of  the  practical  advantages  of  Horner's 
method   is  that  the   first  figure  of  the  root  of  any 


302  ALGEBRA. 

transformed  equation,  after  the  first,  is  in  general 
correctly  obtained  by  dividing  the  last  coefficient 
with  its  sign  changed  by  the  preceding  coefficient, 
which  is  therefore  often  called  the  trial  divisor. 

The    figure    obtained    in    this   way  from    the    first 
transformed  equation  is  likely  to  be  too  large. 

Example.     Find  the  root  of  the  equation 

that  lies  between  3  and  4. 

We  first  transform  equation  (a)  into  another  whose  roots 
shall  be  less  by  3.     The  work  is  given  below. 


-3 

-4 

+  11   [3. 

+  3 

+  0 

j2 

0 

-4 

-     1 

+  3 

+  9 

+  3 

+  5 

+  3 

+  6 

Thus  F(3)  =  —  I,  and  the  first  transformed  equation  is 

^(3  +  ;ri)  =  :ri3  +  6:ri2  +  5  ;ri  -  I  =  0,  (l) 

of  which  the  root  sought  is  positive  and  less  than  i. 

Since  x^<,  \,x^  <  x^  <  x^.  Hence  it  is  probable  that  the 
Jirst  figure  of  this  root  of  (i)  will  be  correctly  given  by  the 
quadratic  equation 


THEORY   OF  EQUATIONS.  303 

We  next  diminish  the  roots  of  (i)  by  o.i. 


+  6.0 

+  5-00 

—  1.000  |o.i 

+    O.I 

+  0.61 

+  0.561 

+  6.1 

+  5.61 

-  0.439 

+  0  I 

+  0.62 

+  6.2 

+  6.23 

+  0.1 

+  6.3 

Thus  /^(3.i)  =  —  0.439,  ^"^  ^^6  second  transformed  equa- 
tion is 

F(3.  i  +  x^)-  x^  +  6.3  x^  +  6.23  ^2  -  0.439  =  0,  (2) 

of  which  the  root  sought  is  positive  and  less  than  o.i. 

Since  x,^^  <  o. i,  x^  and  x.^  are  each  much  smaller  than  .i^. 
It  is  probable  therefore  that  the  first  figure  of  this  root  of  (2) 
will  be  correctly  given  by  the  simple  equation 

6.23  x^  —  o  439  =  0  ;      ''.  X2  =  0.06  +. 
We  next  diminish  the  roots  of  (2)  by  0.06. 

1  +6.30  +6.2300        —0439000  1 0.06 

+  0.06  +0.3816        +0.396696 

+  6.36  +6.6116        -0.042304 

+  006  +  0.3852 

+  6.42  +  6.9968 

+  o  06 
+  6.48 
Thus  7^(3.16)  is  — ,  and  the  third  transformed  equation  is 
/^(3. 16  +  ^a)  =  V  +  6.48  V  +  6.9968  ^3  -  0.042304  =  0,     (3) 
of  which  the  root  sought  is  positive  and  less  than  o.oi. 


304  ALGEBRA. 

Dividing  0.042304  by  6.9968  we  find  the  next  figure  of  the 
root  to  be  o  006.  Diminishing  the  roots  of  (3)  by  0.006  will 
give  the  next  transformed  equation,  which  will  furnish  the  next 
figure  of  the  root ;  and  so  on. 

But,  since  x^  <  o.oi,  and  the  coefficient  of  x^^  is  less  than 
that  of  Tg,  it  is  probable  that  the  first  Iwo  figures  of  x^  will  be 
correctly  given  by  the  simple  equation 

Xq  =  0.042304  -^  6.9968  =  0.0060 -f. 

Hence  3.1660  is  the  required  root  of  (a)  to  four  places  of 
decimals. 

Observe  that  the  known  term  of  any  transformed  equation  is 
the  value  of  J'ix)  for  the  part  of  the  root  thus  far  found. 

413.  As  seen  above,  the  known  term  of  any  trans- 
formed equation  is  the  value  of  F{x)  for  the  part  of 
the  root  thus  far  found ;  hence  the  known  term  must 
have  the  same  sign  in  all  the  transformed  equations. 
If  any  figure  of  the  root  is  taken  too  large,  the 
known  term  in  the  next  equation  will  have  the 
wrong  sign.  If  a  figure  is  taken  too  small,  the  root 
of  the  next  equation  will  evidently  be  of  the  same 
order  of  units. 

Hence,  eack  figure  of  the  root  is  correct  if  the  next 
transformed  equation  has  a  known  term  of  the  same 
sign  as  that  of  the  preceding  equatio7i  and  a  root  of  a 
lower  order  of  units. 

Example.  Find  the  root  oi  x^  —  'i,x'^  —  ^x  -\-  \\  =0  that 
lies  between  i  and  2. 

We  give  below  the  work  of  the  successive  transformations 
written  together  in  the  usunl  form.     The  broken  lines  mark  the 


THEORY   OF  EQUATIONS. 


305 


conclusion  of  each  transformation,  and  the  figures  in  black-letter 
are  the  coefficients  of  the  successive  transformed  equations. 


1.  -3                           -4 
I                           -  2 

4-  II   1  1.782 
-   6 

—  2 
i 

-6 
—  I 

5.000 

-    4-557 

I 

-7.00 

0.49 

0.443000 

—    0.428448 

0.0 

0.7 

-6.51 
0.98 

0.014552000 

—  0.010340232 

0.7 
0.7 
1.4 
0.7 

-  5.5300 

0.1744 

-5.3556 
0.1808 

0.004211768 

2.10 

0.08 

-  5.174800 

4684 

2.18 
0.08 

—  5.1701 16 
4688 

2.26 
0.08 

-  5.165428 

2.340 
2 

2.342 
2 

2.344 
2 

2.346 

Here  we  find  that  the  second  figure  of  the  root  is  correctly 
given  by  dividing  5  by  7;  the  third  by  dividing  0.443  by  5.53  ; 
the  fourth  by  dividing  0.014552  by  5.1748  ;  and  so  on. 


306  ALGEBRA. 

Since  x^<.  o.ooi,  and  the  coefficient  of  x^^  is  much  less  than 
that  of  x^,  it  is  probable  that  the  first  three  figures  of  x^  are 
correctly  given  by  x^  =  0.00421 176S  -^  5.165428  =  0.000815. 
Hence  i. 78281 5  is  the  required  root  to  six  places  of  decimals. 

How  many  figures  of  the  root  will  in  this  way  be  correctly 
given  by  the  last  transformed  equation  may  be  inferred  from  the 
value  of  its  root  and  the  relative  values  of  its  leading  coefficients. 

414.  Negative  Roots.  To  find  a  negative  root,  it  is 
simplest  to  change  the  sign  of  the  roots  (§  405), 
obtain  the  corresponding  positive  root,  and  change 
its  sign. 

Thus  to  find  the  root  of  .r^  —  3  ;ir2  —  4  ;r  +  11  =  0  that  lies  be- 
tween —  I  and  —  2,  we  obtain  the  positive  root  of  the  equation 

x^  -\-  2  x^  -  4 X  —  II  =  0  §  405. 

that  lies  between  i  and  2,  and  change  its  sign. 

It  is  evident  that  Horner's  method  is  directly  ap- 
plicable to  an  equation  in  which  the  coefficient  of 
x"  is  not  unity. 

Note.  For  a  fuller  discussion  of  Horner's  method,  for  its 
application  to  cases  where  roots  are  very  nearly  equal,  and  for 
contractions  of  the  work,  see  Burnside  and  Panton's  or  Tod- 
hunter's  "Theory  of  Equations." 

EXERCISE   60. 

Find  to  five  places  of  decimals  the  root  of  the  equation 

1.  x^  +  X  —  500  =  0,  that  is  7  +. 

2.  x^  —  2  X  —  ^  =  0,  that  is  2  +. 

3.  x^  —  sx^  +  Sx—  I  =0,  that  is  0  +. 


THEORY   OF  EQUATIONS.  307 

4.  x^  +  2  X-  —  ^  X  —  g  =  0,  that  is  i  +. 

5.  2  x^  +  $x  -^  go  =  0,  that  is  3  +. 

6.  x^  -[■  x^  +  Tox  -{-  300  —  0,  that  is  —  (3  +). 

7.  5  ^^  -  6  a:^  4-  3-^  +  85  =  0,  that  is  -  (2  +)• 

8.  sx'  -\-  x^-{-  4x^+  s^-  375  =  0,  that  is  3  +. 

9.  a*  —  80  Jt'  +  1998  a:^  —  14937  X  +  5000  =  0,  that  lies 
between  30  and  33  ;  between  ;^;^  and  40. 

Find  the  positive  root  of  each  of  the  equations : 

10.  2x»  — 85  jc^  — 85:^-87  =  0. 

11.  4:<;3—  13.V2  — 3i.r  —  275  =  0. 

12.  20 a:^ —  121^^—  121. r —  141  =0. 

13.  Solve  a;^  —  315  Ji'"^  —  19684^:1;  +  2977260  =  0. 

Reciprocal  Equations. 

415.  A  Reciprocaly  or  Recurring,  equation  is  one 
that  remains  unaltered  when  x  is  changed  into  its 
reciprocal;  that  is,  when  the  coefficients  are  written 
in  reverse  order  (§  406).  Hence  the  reciprocal  of 
any  root  of  a  reciprocal  equation  is  also  a  root. 

Therefore,  if  the  equation  is  of  an  odd  degree, 
one  root  is  its  own  reciprocal  ;  hence  one  root  of  a 
reciprocal  equation  of  an  odd  degree  is  +  i  or  —  i. 


308  ALGEBRA. 

Example.     Given  5  as  one  root,  to  solve  the  equation 

5;ir5  —  51  jr^  +  i6o;r3  —  l6o;if2  +  51  ;»;  —  5  =  0.       (i^ 

Since  (i)  is  a  reciprocal  equation,  and  one  root  is  5,  a  second 
root  is  \.  Since  (i)  is  of  an  odd  degree,  one  root  is  +  i  or  —  i. 
By  inspection  we  see  that  +  i  is  a  root. 

The  depressed  equation  is 

x^-  4X+  1=0, 

I 


from  which  x  =  2  +  x/ 3  or  2  —  a/ 3,  which  equals  ,~ 

2  +  V3 

A  reciprocal  equation  of  an  odd  degree  can  be 
depressed  to  one  of  an  even  degree;  for  one  of  its 
roots  is  always  known. 

416.  Let^'^+A^"-  '+/2^"-'+---+A-i^+A=^.  (0 
and    A^"+A-i^""'+A-2^"~'+---  +  A-^"+i  =  0,   (2) 

be  equivalent  equations;  that  is,  let  (i)  be  a  re- 
ciprocal equation.  Dividing  (2)  by  A  to  put  it  in 
the  form  of  (i),  and  then  equating  their  last  terms, 
we  have 

A=  I  -^A'   •••  A  =  ±  I- 

Reciprocal  equations  are  divided  into  two  classes, 
according  as  A  is  +  i   or  —  i. 

Ftrs^  Class.     \{ pn  =  i ;   then,  from  (i)  and  (2), 

P\—Pn-\,    A=A-2V*--^ 

that  is,  the  coefficients  of  terms  equidista7it  from  the 
ends  of  F  (x)  are  equal. 


THEORV   OF  EQUATIONS.  309 

Second  Class.     If /„  =  —  i ;    then,  from  (i)  and 

(2), 

P\  =  —  A-ij  A  =— A-2J  -"'f 

that  is,  t/ie  coefficients  of  terms  equidistant  from  the 
ends  of  F  (;r)  are  equal  numerically  but  opposite  in 
sign. 

If  in  this  class  n  is  even,  say  «  =  2  m ;  then  p,^ 
—  —pmy  or  p,„  =  0)  that  is,  the  middle  term  is 
wanting. 

417.    Standard  Form.     Any  equation  of  the  second 

class  of  even  degree  can  evidently  be  written  in  the 

form 

^"  -  I  -\-p,x{x--^  -  i)  +  ...  =  0.  (i) 

Since  n  is  even,  F(x)  in  (i)  is  divisible  by  ;r2—  i  ; 
hence  two  roots  of  (i)  are  ±  i. 

The  depressed  equation  will  evidently  be  a  recip- 
rocal equation  of  the  first  class  of  even  degree,  which 
is  called  the  standard for7n  of  reciprocal  equations. 

Hence,  afty  reciprocal  equation  is  in  the  standard 
form  or  can  be  depressed  to  that  form. 

418  Any  reciprocal  equation  of  the  standard  form 
can  be  reduced  to  one  of  half  its  degree.  The  fol- 
lowing example  will  illustrate  this  truth. 

Example.     Solve  ;r*  —  5  ;r8  +  6.r2  —  5 :r  +  i  =  0.  • 
Dividing  by  (;f*)2,  or  x^^  we  obtain 


310  ALGEBRA. 

Since  x^  +  -^=  Ix  +  ~]   —  2,  from  (i)  we  have 


;ir  +  -  =  4  or  I  ; 


.-    I   ±   a/—  3 

...   ;r=2±  V3, ^ 


EXERCISE   61. 

Solve  the  following  equations  : 

1.  x'^  +  x^  +  x^  +  x'^  +  X  +  1  =0, 

2.  X*  —  ^  x^  -\-  ^  X  —  I  =  0. 

3.  X*  —  lojc^  +  26:^:^  —  lojt;  +  I  =  0. 

^.  x^  —  ^  x"^  +  g  x^  —  c)  X-  -{-  ^  X  —  I  =^  0. 

5.  x^  +  2  x^  —  ;^  x^  —  ;^  x"^  +  2  X  +  I  =  0, 

6.  ^^  —  ^^  4-  ^^  —  -^^  +  ^  —  I  =  0. 

7.  6  Jt:"  +  5  ^-^  —  38  Jt^  +  5  .r  +  6  =  0. 

8.  ^»  — /7^2  +  /^jt:—  I  =0. 

419.    Binomial  Equations.      The    two   general   forms 
of  Binomial  Equations  are 

x^  —  d  =  0,  (i) 

and  x"-{-d  =  0,  {2) 

in  which  d  is  any  positive  number. 


THEORY   OF  EQUATIONS.  3H 

By  §  405,  when  n  is  odd,  the  roots  of  (2)  are  the 
roots  of  (^i)  with  their  signs  chajiged. 

The  n  roots  of  either  (i)  or  (2)  are  unequal;  for 
.r"  q:  b  and  its  derivative  «^"~'  have  no  C  D. 

From  (i),  ;r  =  "^Z  b ;  from  (2),  x  =  y—  b  ;  that  is, 
each  of  the  n  unequal  roots  of  (i)  or  (2)  is  an  «th 
root  of  +  ^  or  —  ^. 

Hence,  anj/  number  has  n  tmequal  nth  roots. 

By  §  391 »  the  n  roots  of  ;r"  —  i  =  0  multiplied  by 
y  a  are  equal  to  the  n  roots  of 

X"  —  a  =  0. 

Hence,  «//  //;^  nth  roots  of  any  number  may  be 
obtained  by  multiplying  any  one  of  them  by  the  nth 
roots  of  unity. 

If  n  is  even,  ;r"  —  i  =  0  has  two  real  roots,  ±  i. 

If  «  is  even,  x"  ■\-  i  =  0  has  no  real  root;  for  ^J  —\ 
is  imaginary  when  71  is  even. 

If  n  is  odd,  ;ir"  —  i  =  0  has  one  real  root,  +  I ;  and 
;r"  +  I  =  0  has  one  real  root,  —  i. 

420.    The  Cube  Roots  of  Unity.      The  roots  of 

A:«  -  I   =   (X  -  l)   (^2  ^  AT  +   i)   =  0, 

were  found  to  be 

I,  -  ^  +  \  V"^,  -\-\  \/^. 

If  ft)  denote  either  of  the  imaginary  roots,  by 
actually  squaring,  the  other  is  found  to  be  d?. 
Hence  the  three  cube  roots  of  +  i  are  i,  ft),  and  ft)^. 


312  ALGEBRA. 

Therefore  by  §  419  the  three  roots  of  ^"'^  +  i  =  0,  or 
the  three  cube  roots  of  —  i,  are  —  i,  —  w,  and  —  w^. 

Example.    Find  the  five  fifih  roots  of  32  and  —  32. 
The  equation  ;r5  =  i  is  equivalent  to  ;ir  —  i  =  0 
and  x^  +  x^  +  x^  +  x+  1  =0.  (i) 

2 


From  (I),  ^^+i)    +  ^;ir  +  -|,j  =  i; 


...    ■.  + 1  =  ^14^5. 

X  2 

Solving  ;ir  —  I  =  0  and  the  two  quadratics  in  (2),  and  multi- 
plying each  root  by  /y^32,  or  2,  we  find  the  five  fifth  roots  of 
32  to  be  2, 


-  I  +  Vs  ±  V-  10  -  2  V5    -  I  -  V5  ±  V-  10  +  2  V5 

■  2  '  2 

These  roots  with  their  signs  changed  are  the  roots  of  -  32. 

*421.    Solution  of  Cubic    Equations.      By   §410    the 
general  cubic  equation 

can  be  transformed  into  another  of  the  simpler  form 
x^+  7,Ifx+  0  =  0.  (i) 

To  solve  this  equation,  assume 

xz=r^  -{-  s^  ;  (2) 

,-.   x^  —  3r^s^x-'(r  +  s)  =  0.  (3) 


THEORY   OF  EQUATIONS.  313 

Comparing  coefficients  in  (i)  and  (3),  we  have 

r^s^  =  -Jjr,r  +  s  =  -G,  (4) 

Solving  these  equations,  we  obtain 


r=},{-G+VG'  +  4^%  (5) 


Substituting  in  (2)  the  value  of  s^  obtained  from 
the  first  of  equations  (4),  we  have 

^  =  rh  +  ^,  (6) 

the  value  of  r  being  given  in  (5). 

If  ^  r  denote  any  one  of  the  cube  roots  of  r,  by 
§419,  rS"  will  have  the  three  values,  v^r,  w  v^r,  w^  Vr, 
and  the  three  roots  of  (i)  will  be 

3         -If        3/-   ,    -If      ,  3/-   ,    -B- 

If  in  (6)  we  replace  r  by  s,  the  values  of  ;i:  will  not  be 
changed  ;  for,  by  (2)  and  the  relation  t^  s^  =  ~  H, 
the  terms  are  then  simply  interchanged.  Moreover, 
the  other  solution  of  equations  (4)  would  evidently 
repeat  these  values  of  ;r. 

Note.  The  above  solution  is  generally  known  as  Cardan's 
Solution,  as  it  was  first  published  by  him,  in  1545.  Cardan 
obtained  it  from  Tartaglia ;  but  it  was  originally  due  to  Scip'o 
Ferreo.  about  1505.  See  historical  note  at  th^  end  of  Burnside 
and  Panton's  "Theory  of  Equations," 


314  ALGEBRA. 

*  422.  Application  to  Numerical  Equations.  When  a 
numerical  cubic  has  a  pair  of  imaginary  or  equal 
roots,  by  Example  3  of  §  403,  6^^  _|_  ^  //3  ->  ^j.  _  q. 
hence  r  in  (5)  of  §  421  is  real,  and  therefore  the 
roots  may  be  computed  by  the  formula  (6). 

When,  however,  the  roots  of  a  cubic  are  all  real 
and  unequal,  by  Example  3  of  §  403,  G^  -\-  ^H^ <  0; 
whence  r  is  imaginary,  and  the  roots  involve  the  cube 
root  of  a  complex  number.  Hence,  as  there  is  no 
general  arithmetical  method  of  extracting  the  cube 
root  of  a  complex  number,  the  formula  is  useless  for 
purposes  of  arithmetical  calculation.  In  this  case, 
however,  the  roots  may  be  computed  by  methods 
involving  Trigonometry. 

When  the  real  root  of  a  cubic  has  been  found  by 
(6)  or  (2)  of  §  421  it  is  simpler  to  find  the  other  two 
roots  from  the  depressed  equation. 

Example.        Solve  ;ir^  —  15  ^  —  126  =:  0.  (i) 

Put  x=r^  +  s^', 

.'.     x^-^r^s^  X-  (r  +  s)  =  0]  (2) 

r^ s^  =  5,  r  +  s  =  126; 
r^  =  5,   s^  =  I ; 

x=  r^  +  s^  =  S  +  T  =6, 
Hence  the  depressed  equation  is 

x^  +  6x+  21  =0, 
and  the  three  roots  are  6,    —3  +  2  /y/^^    —  3  —  2  v^'s* 


THEORY   OF  EQUATIONS.  315 

EXERCISE   62. 

1.  Find  the  6  sixth  roots  of  729  ;  of—  729. 

2.  Find  the  8  eighth  roots  of  256  ;  of—  256. 

Solve  the  following  equations  : 

3.  x^—iSx  =  ^^.  6.   ^' +  21  X  =  —  342. 

4.  :x:3  +  63^  =  316.  7.    .T»  + 3.t'^  +  9.r  =  13. 

5.  x^  -\-  'J  2  X  —  J  J  20.  8.   x^  —  6  a-  +  3.r  =  18. 
9.   x^  —  6x-  +  J^x  =  10. 

10.   x^  —  15  .^^  —  33  'V  =  —  847. 

*  423.    Solution  of  Biquadratic  Equations.       Any    bi- 
quadratic equation   can  be  put  in  the  form 

X*  +  2px^  +  qx"^  -\-  2rx  ^  s-=  0.  (i) 

Adding  {ax  +  ^)^  to  both  members,  we  obtain 

a:*  ^  2px^  -\-  {q  +  a-)  x^  +  2  (r  +  ^7  /»)  x 

+  s  +  l,'  =  (ax  +  l,y,  (2) 

Assume 

X*  +  2p  x^  +  (q  +  a^)  x^  +  2  (r  +  a  li)  X 

+  s  +  P  =  (x'+/>x  +  ^y,      (3) 

Equating  coefficients  (§  262),  we  have 

P^+  2k  =  q  +  a\  />k  =  r  +  ad,    J^^  =  s  +  P.      (4) 


3l6  ALGEBRA. 

Eliminating  a  and  b  from  (4),  we  have 

or     2k^  —  q  k"^  -{-  2  {p  r  —  s)  k  —  p'^  s  -^  q  s  —  r'^  —  ^, 

From  this  cubic,  find  a  real  value  of  k  (§  389). 
The  values  of  a  and  b  are  then  known  from  (4). 
Subtracting  (2)   from  (3),  we  have 

{x^-^rpx  -^  kf-{ax  +  bf  =  0, 

which  is  equivalent  to  the  two  quadratic  equations 

x^  +  (p  —  a)  X  +  {k  —  Z*)  =  0, 
and  ^2  ^  (/  +  «)  ^  +  (^  +  ^)  =  0, 

of  which  the  roots  are  readily  obtained. 

Note.  The  solution  given  above  is  that  of  Ferrari,  a  pupil 
of  Cardan.  This  and  those  of  Descartes,  Simpson,  Euler,  and 
others,  all  depend  upon  the  solution  of  a  cubic  by  Cardan's 
method,  and  will  of  course  fail  when  that  fails.  For  a  full 
discussion  of  Reciprocal,  Cubic,  and  Biquadratic  Equations 
consult  Burnside  and  Panton's  "Theory  of  Equations." 

Example.   Solve  x^  —  6  x^  ^  \2  x'^  —  14  ;ir  +  3  =  0. 
Adding  (ax  -\-  by-  to  both  members,  we  obtain 

r4-6jr3+(i2  +  «2);t-2+2(^^-7)^  +  ^^  +  3  =  («-^+'^)^.      (0 
Since  ^  =  —  3,  assume 

or*  —  6  ;jr3  +  (12  +  «2)  ^2  _f_  2  («  3  _  7)  ;r 

+  ^-^  +  3  =  (^'-3^  +  >C02;        (2) 
.♦.    12  +  ^2^9+ 2/^,   ab-T=-zk,   ^2  +  3^^2. 

...     y&3_6^2+   18k  -20  =  0. 


THEORY   OF   EQUATIONS.  317 

Whence  ^  =  2;  hence  d^  =  \^  b"^  =  i^  ab  =  i.  (3) 

From  (i),  (2),  and  (3),  we  obtain 

(;r2  -  3  ;r  +  2)2  ~  {x  +  1)2  z=  0, 
which  is  equivalent  to  the  two  equations 

;r2  -  4;ir  +  I  =  0,    ;i-2  -2 ^  +  3  =  0  ;  « 

.'.  x=2±^/]„  I  ±  V^-^. 

EXERCISE  63. 

Solve  the  following  equations  : 

1.  x^  ^  ^x^ -\-  ()x^^2>x—  10  =  0, 

2.  x^  —  3  r^  —  42  jc  —  40  =  0. 

3.  X^  +    2X^—  7^:^  —  8^+    12=0. 

4.  x^  —  2>^'^  —  ^^  —  2  =  0. 

5.  x^  —  1 4  ^'  +  59  :*r2  —  5o  X  —  36  =  0. 
t 

6.  x^—2x^—  12 x"^  +  io.r  +  3  =  0. 

7.  :«:*  —  2  .;c^  —  5  *•-  +  10  a:  —  3  =  0. 


3i§ 


ALGEBRA. 


CHAPTER    XXII. 

DETERMINANTS. 


424. 

Determinants  of  the  Second  Order. 

The  square 

array 

«2       h 

(') 

is  called  a  determiitant  of  the  second  order. 

The  quantities  a\,  a^,  bu  b^  are  called  Elements. 

A  horizontal  line  of  elements,  as  a^,  bu  is  called  a 
Row ;  and  a  vertical  line,  as  ^i,  a^,  a.  Column.  The 
Order  of  a  determinant  is  determined  by  the  number 
of  elements  in  a  row  or  column. 

The  downward  diagonal,  a^  b^,  is  called  the  Princi- 
pal Diagonal ;  and  the  upward  diagonal,  a<i  b^,  the 
Secondary  Diagonal. 

The  determinant  (i)  stands  for  the  expression 
^1  ^2  ~  ^2  biy  which  is  called  the  Expansion  of  this 
determiftant. 

Thus  the  expansion  of  a  determinant  of  the  second  order  is 
the  product  of  the  elements  in  the  principal  diagonal  minus  the 
product  of  the  elements  in  the  secondary  diagonal.     Hence 


-4    -3 
7    -5 

5    7 
2     6 


=  (-4)(-5)-7(-3)E4i. 


=  44. 


=  -8. 


DETERMINANTS. 


3^9 


EXERCISE  64. 

Expand  the  determinant 

I. 

I       X     . 

3- 

x-vy 

X 

X       1 

x-y 

-y 

2. 

0    ->& 

, 

4-    ^-y 

x^- 

c          b 

X  +y 

^'M 

xy  ^  f 


5.   Write  as  a  determinant  ax  —  by;  as  -\-bk;  ax  -\-  by; 
axb  +  cdy;  x^  -\- y^;  c—  s ;  a  +  b;  ax. 


Show  by  expanding  that 


^1 

«1 

=  0; 

ai 

h 

«2 

a2 

di 

b. 

=  0. 


I.  That  is,  if  the  corresponding  elements  of  two  col- 
umns or  tivo  rows  of  a  determinant  are  alike ^  the 
deter^ninant  equals  zero. 


7- 


a^x 

C\ 

=.  X 

a-i 

Cx 

= 

a<iX 

^2 

^2 

^2 

a^x    CtX 
a<i  C2 


II.  That  is,  if  each  element  of  a  column y  or  row,  be 
multiplied  by  a  given  fiumbery  the  determinant  will  be 
multiplied  by  that  number. 


a^  +  bx     c^ 
a^  +  b<i     C2 


b\     Ci 
^2     ^2 


ai  -\-  bi     rt-a  -f-  bg 

Ci  Ci 


320 


ALGEBRA. 


III.  That  is,  if  each  element  of  any  column,  or 
row,  is  the  sum  of  two  or  more  numbers,  the  deter- 
minant can  be  expressed  as  the  sum  of  two  or  more 
determinants. 

Note.  The  properties  I.,  II.,  III.  are  fundamental  in  solv- 
ing systems  of  linear  equations  by  determinants.  Their  general 
proof  will  be  found  in  §§  441,  445,  and  446. 


425.    To  solve  by  determinants  the  system 
\)S^\a^x  +  b^y  =  rn^,     (i) 


a<^x  +  b^y  =  m^.     (2) 


{a) 


Multiplying  (i)  by  b^,  and  (2)  by  —bx,  and  adding 
the  resulting  equations,  we  obtain 

Wi^2  —  W2<^i  =  {a^x  +  b^y)  b^  —  {a^x-^-  b^y)  b^. 


or 


^1     bi 

= 

ai  X 

\b 

iy 

i 

1 

(3) 

m^     ^2 

azX 

+  b^y     be, 

= 

ai  X 
a^x 

bx 
b. 

+ 

bxy 
biy 

by  III. 

=  X 

ax 
^2 

bx 
b. 

+y 

b. 

by  II. 

=iX 

ax 

«2 

bx 
b. 

' 

by  I. 

/.  :ji 

?  = 

nx 

'/2 

bx 
b. 

4- 

^1 
^2 

b. 

(4) 

DETERMINANTS. 


321 


Again,  multiplying  (2)  by  ^1,  and  (i)  by 
adding,  we  obtain 


The  determinant 


ai 

aix  +  b^y 

= 

y 

a-i 

h 

«2 

cit  X  +  b^y 

Ci 

b'. 

ai 

wi 

-H 

«i     ^1 

^2 

in<i 

^2      ^2 

• 

a\     K 

^2 

^2 

> 

a^,  and 

(5) 

(6) 


whose  elements  are  the  coefficients  of  the  unknowns 
in  system  (a)  written  in  their  order,  is  called  the 
Determinant  of  the   System. 

Hence,  the  value  of  either  unknown  is  a  fraction 
whose  denominator  is  the  determinant  of  the  system ^ 
and  zvhose  mimerator  is  the  deterntinant  of  the  system 
with  the  k7iown  terms  substituted  for  the  coefficients  of 
the  unknown  whose  value  V5  sought. 

Note.  It  was  the  study  of  systems  of  linear  equations  that 
suggested  the  idea  of  determinants  and  their  notation.  The  next 
few  pages  are  designed  to  illustrate  the  beauty  and  utility  of  de- 
terminants in  the  soludon  of  such  systems. 

Example.     Solve  the  system        zx  — 

7^  + 
Writing  the  given  system  in  the  form  of  system  {a),  we  liave 

2^-7  =  9.  ) 
3^-7j=  19.  f 

4;     y^ =_I. 


f  19  =  3  a-.') 


9 

—  1 

19 

-7 

2 

-I 

3 

-7 

2 

9 

3 

19 

2 

—  I 

3 

-7 

322 


ALGEBRA. 


EXERCISE    65. 


Solve  the  system 

1.  sx-\-6y=  17, 
6x  +sy  =  16. 

2.  Sx—    y^S4, 

X  +  8y  =  53. 

3.  ^x='jy-2j,l 
2JX  =  gy  +  75.  i 

12  a:  =  9_>' 

5-  >^^+  ^^J'^i 


6.  :v  +  7   =  rt;  +  ^,    j 
ax  —  dy  =  P  —  d^.  \ 

7.  ;«:  —    ^    =  jj;  _  ^^ 
«  ^  —  2  ^"  ==  /^_);  —  2  ^ 

8.  bx—  b^  =ay,  ) 
ax  —  by  —  a^.    \ 


9.    a^x  ■\-  b^y  =  c^, 
a^x  +  b^y  =  ^. 
10.    ax  +  by  —  a^  +  z^^, 
,'o>'  — 3-^  bx-\-ay  =  2ab. 

II.    (^  +  /^)  Jt:  —  (rt!  —  Z') _)/  =  3  rt;  ^, 
(rt;  +  /;)  J*  —  (rt;  —  /^)  j:t:  =  rtt  <^. 

Note.  The  student  should  now  solve  the  preceding  exam- 
ples mentally^  without  writing  out  the  values  of  x  and  y  in  the 
determinant  form. 

426.    Determinants  of  the  Third  Order.    The  ordinary 
form  of  a  determinant  of  the  third  order  is  that  in  (i), 
h       , 

h         ^2  (l)  ^72        \h        '^2        (2) 


and  its  expansion,  or  development,  is 

^1  K  ^3    +  ^2  ^3  ^1   +   ^3  '^l  ^^2  —  ^3  <^2  <^i   —  ^2  '^i  '^3  —  ^1  ^2,  ^2- 

Thus,  the  expansion  of  a  determinant  of  the  third 
order  is  the  sum  of  the  three  products  obtained  from 
the  principal  diagonal  and  lines  parallel  thereto,  as 


DETERMINANTS. 


323 


indicated  in  form  (2),  minus  the  sum  of  the  three 
products  obtained  from  the  secondary  diagonal  and 
lines  parallel  thereto.  Each  product  includes  three 
elements,  one,  and  only  one,  from  each  column  and 
each  row.  The  term  a^  ^2.  ^3  is  called  the  Principal 
Term. 

In  the  above  notation  the  number  of  the  column 
to  which  a  given  element  belongs  is  indicated  by  the 
letter,  and  that  of  the  row  by  the  subscript  number; 
thus,  the  element  b.^  belongs  to  the  second  column 
and  the  third  row. 


According  to  this  notation  we  have 


X    —m    s 

y       n     t 

Z      —  O       71 


^z:  xnu  —  y OS  —  z JH t  —  z ns  ■{- y  mu  -\-  xo t. 


2     - 1     —4 
3-35 


=  -  5  +  24  -  24  -}-  12  -I-  20  -  12  =  15. 


Evaluate 


EXERCISE    66. 


I      2     3 

4- 

4 

—  I 

—  2 

7- 

X 

y  z 

2    3    4 

0 

3        0 

V 

w   u 

3   4   5 

3 

-7        4 

t 

r    s 

4        5 

2 

5- 

I    —  I         I 

8. 

^1 

a^  a^ 

—  I        2 

3 

3        I    -4 

K 

K   b„ 

6  -^ 

5 

2    -3   -5 

^1 

C.2     c^ 

I  —I 

I 

6. 

I 

\   \ 

9- 

a 

~-2a 

b 

4  -3 

0 

\ 

\   \ 

3* 

—  c 

Ad 

3       2 

- 

-5 

\ 

\    \ 

2C 

: 

5^ 

-4^ 

324 


ALGEBRA. 


lo.  Write  as  a  determinant  each  of  the  following  expres- 
sions : 

(i)  ams  —  dmx  —  els  —  atr  -\-  cix  +  dlr ; 

(ii)  aks  —  rka  +  sma  +  r^ I  —  s^l  —  amr ; 
(iii)  r^  —  k'^  -\-  2srx  —  2sxk, 

Show  by  expansion  that 


II. 


12. 


13- 


^2        ^2 


=  0, 


a. 

^ 

^1 

cii 

^1 

^1 

^2 

h 

^2 

=  0. 


ma-^ 

b. 

Cl 

—  in 

ma^ 

Ih 

Ci 

ma^ 

K 

'z 

«i  +  h^ 

^2  +    bo 


^2 


«2  ^2 


^2 


+ 


^1        ^1 
^2        ^2 


^1  +   '^l       ^2  +   ^2 

Note.     Examples  ii,  12,  13  prove  for  determinants  of  the 
third  order  the  principles  I.,  II.,  III.  of  Exercise  64. 


«3  +  ^3 
^3 


427.  A  determinant  whose  elements  are  expressed 
by  letters  with  stibscripts  is  often  denoted  by  writing 
between  bars  its  principal  term,  from  which  all  its 
other  terms  are  known.  Thus,  the  identities  in  Exam- 
ples 12  and  13  of  Exercise  66  may  be  written  briefly 
as  follows : 

\max     b^     c^\  =  m\ai     b^    c^\, 


and     I  ^1  +  /^i 


4  I  =  I  ^i     ^1    4  1  +  I  ^1     ^2    ^3 


DETERMINANTS. 


325 


428.    To  solve  by  determinants  the  system 


ax  X  +  /?iy  +  CiZ  =  mi, 
a^x  ^  b.^y  +  C2Z  =  m^, 
a^x  +  b.^y  +  CzZ  =  m^, 


{a) 


In  §425  the  first  determinant  of  equation  (3)  or  (5) 
may  be  obtained  from  the  determinant  of  the  sys- 
tem by  writing  the  known  terms  in  place  of  the 
coefficients  of  the  unknown  whose  value  is  sought; 
and  the  second  may  be  obtained  from  the  first  by 
putting  for  the  known  terms  their  values  as  given  in 
(i)  and  (2).  Writing  equation  (4)  below  according 
to  this  law,  we  find  the  value  of  x  as  follows ; 


wi     bi 

^1 

= 

aix  -\-  biy  ■ 

■{-  CiZ     bi     Cx 

m^     b^ 

^2 

a,x  +  b^y  ■\-  c^z     b^    c^ 

(4) 

nt^     ^8 

^3 

H^-^b^y^c^z     ^3    ^3 

= 

ax  X    b^    Cx 

+ 

b^y    bx    Ci 

+ 

c^z    bi 

Cx 

a^x    b^    c^ 

biy    b^    Ci 

c^z    bi 

^2 

a^x     b^    ^3 

bzy    b^    ^3 

^3^    h 

^3 

=  ^ki    bi   c^\^-y\bi    b^    rgl  +  ^ki    ^2 

^al 

= 

x\ai    b^    c^\ 

x  =  \mx    b^    ^sl-^l^i    ^2    h 


Similarly  we  obtain 


and 


y  =  \a^     m^      c.\~\ax     b^    c^\^ 
z  =  \ax     b^     W3  \-^\ax     b^     c^\  . 


326 


ALGEBRA. 


Hence,  in  a  system  of  three  linear  equations,  the 
value  of  each  unknown  may  be  written  out  according 
to  the  rule  in  §  425. 


Example.     Solve  the  system 


x  = 


2    —2 
5        3     - 


-2 
3     - 


3  .r  -  2  J  H-  2-  =  2 

2  X  +  3/  -  2"  =  5 

X  +  y  -\-  z  —  6 


y 


■\ 


3 

2         I 

2 

5    -I 

I 

6        I 

3 

—  2        I 

2 

3        I 

I 

I         I 

=  2,  ^  =  3. 


EXERCISE    67. 


Solve  the  system 


X 
2  X 
3^ 


2j|;  +  32:1=:  2, 

3j>;  +    2     =:  I, 

^    +22  =  9. 


2.  ^+J^'+2:=:I,^ 
2^+3^  +  2:=  4,  |- 
4  ji;  +  9j/  +  2:  —  16.  J 

3.  ^+ 2j>;  — 32=  6,  \ 
2^  +  4_y— 72:  =  9,  |- 
3^-  y   — 5^  =  8.  J 


:r 

+ 

27 

+ 

2  <3r 

= 

II 

2  JC 

+ 

y 

+ 

2: 

= 

7 

3-^ 

+ 

4y 

+ 

2 

= 

14 

5.  X  +  sy  +  4^=  14, 

Ji:  4-  27  +    2;    =     7, 
2^+     J/     +20=      2. 

6.  «:v  +  dy  =  ij  ^ 

^j;  +r2:  =  I,  |- 
^2:    +  ax  =  I.  -^ 

7.  <r7  +  /^  s:  =  /^  r,  >v 
rtts  -{-  ex  =  c a,  y 
b  X  -\-  ay=^  ab.  ) 

8.  x—ly=  6,\ 
y  -\z^  8,  > 
z  —  \x=^  10.  ^ 


DETERMINANTS. 


327 


In  examples  9  and  10,  consider  the  reciprocals  of  x,y,  z  as  the 
unknowns. 


9- 

I     <      I      1      ^             c 

X      y       z 

10.    I 


+  4=0, 


X     y 

Z        X 


429.  Assuming  that  determinants  of  the  fourth 
order  have  the  properties  I.,  II.,  III.  of  Exercise  64, 
we  may  apply  the  method  of  §  428  to  solve  the  fol- 
lowing system  : 

aix  +  l>iy  +  c^z  -\-  d-iW  —  m^y 

«2 •^"  +  ^2 >'  +  ^t^  -\-  d^w  =  m2, 

a^x  ■\-  b^y  +  qz  ■]-  d^w  =  m^, 

a^x  +  b^y  ■\-  c^z  -\-  d^w  =  vi^. 


-A 


( 

(3)1 

(4) 


(«) 


Wi    b^    ri    //, 

^ 

^1  ^  +  <^i  JJ'  +  ^1 2:  +  ^1  «^    ^1    ^1    dx 

W2    ^2    ^2    ^2 

a^x  -\-  b^y  -\-  c^z  ■{-  d^w    ^2    ^2    ^4 

^3    ^3    ^3    ^; 

«8^  +  ^3^  +  ^8  2  +  ^^3  ^^      ^3     ^3     ^3 

W4     ^4     ^4     ^4 

rt;^  X  +  ^4 JV  +  ^4^  +  ^4  ^      ^4      ^4      ^4 

=  . 

x\ax     b^     c^     d^\ 

,'.  x  =  \  w,    /^2    ^3    <  I  -^  I  ^1    h    ^3    '^rj  .  (5) 

Finding  the  values  of  j,  5-,  and  w  is  left  as  an  exercise  for 
the  student.  The  work  should  be  written  out  in  full,  applying 
only  one  principle  at  a  time. 

The  expansion  of  each  of  the  determinants  in  (5)  contains 
twenty-four  terms.  Similarly  the  solution  of  a  system  of  five 
linear  equations  gives  rise  to  determinants  of  the  fifth  order. 


328  ALGEBRA. 

The  expansion  of  a  determinant  of  this  order  contains  one  hun- 
dred and  twenty  terms.  Thus  we  see  that,  by  the  determinant 
method  of  notation,  systems  of  linear  equations  are  readily 
solved,  and  large  expressions  are  written  in  concise  forms. 

Determinants  of  the  htu  Order. 

430.  Inversions  of  Order.  The  number  of  inver- 
sions in  a  series  of  integers  is  the  number  of  times 
a  larger  number  precedes  a  smaller.  In  the  determi- 
nant (i)  of  §  426,  the  letters  and  the  subscripts  in  the 
principal  term,  ^1  ^2  c^,  are  in  the  natural  order.  If 
the  letters,  or  the  subscripts,  are  taken  in  any  other 
order,  there  will  be  one  or  more  inversions  of  order. 

Thus,  in  the  series  2,  4,  i,  3,  there  are  three  inversions;  2  pre- 
cedes I,  and  4  precedes  both  i  and  3.  In  the  series  <r,  b,  d,  a, 
there  are  four  inversions  ;  c  precedes  both  b  and  a,  b  precedes  a, 
and  ^precedes  a.  , 

431.  If  in  any  series  of  integers  {or  letters)  two  ad- 
jacent integers  {or  letters)  be  interchanged,  the  number 
of  inversions  ivill  be  either  increased  or  diminished  by 
one. 

For  example,  consider  the  two  series  (i)  and  (2), 

3»  7>  ij  6,  5,  8,  2,  4,  9,  (i) 

3>  7»  i»  6,  8,  5,  2,  4,  9.  (2) 

Neglecting  the  relations  of  5  and  8  to  each  other, 
the  inversions  of  one  series  are  evidently  the  same  as 
those  of  the  other.  Interchanging  5  and  8  in  series 
(i),  giving  series  (2),  will  increase,  and  interchanging 


DETERMINANTS.  329 

5  and  8  in  series  (2),  giving  series  (i),  will  decrease, 
the  number  of  inversions  in  the  series  by  one. 

432.  A  product  of  eleinc7its  is  said  to  be  even  or  odd 
according  as  the  total  number  of  inversions  of  both 
its  letters  and  its  subscript  numbers  is  even  or  odd. 

Thus,  the  product  aiC<2,b^  is  even,  while  the  product  b^c^ai  is 
odd. 

433.  The  character  of  a  product  as  even  or  odd  is 
not  cJiaiiged  by  changing  the  order  of  its  elements. 

For  by  the  interchange  of  any  two  adjacent  ele- 
ments in  a  product,  as  a^  c^  b^  d^,  the  number  of  inver- 
sions of  the  letters  is  changed  by  one  (§  431),  so  also 
is  the  number  of  inversions  of  the  subscripts;  hence 
the  total  number  of  inversions  in  the  product  is 
changed  by  an  even  number,  that  is,  by  2  or  0. 
Hence  an  even  (or  odd)  product  will  remain  even 
(or  odd),  whatever  change  be  made  in  the  order  of 
its  elements. 

Thus,  the  product  d^  a.^  c^  bi  is  even,  if  a^  b^  c^  d^  is  even. 

434.  Hence  the  simplest  method  to  determine 
whether  a  product  is  even  or  odd  is  to  arrange  its 
elements  in  the  natural  order  of  the  letters,  and  then 
count  the  number  of  inversions  of  the  subscripts  ;  or  we 
might  arrange  the  elements  in  the  natural  order  of  the 
subscripts^  and  theft  count  the  number  of  inversiojis  of 
the  letters. 

Thus,  the  product  b^  d^ ^i  ^^2  is  even,  since  a^  b^  Ci  d^  or  Ci  a.^ b-i d^ 
is  even. 


330  ALGEBRA. 

EXERCISE    68. 

Show  that  the  product 

1.  <7o  b^  Ci  </^  is  even.  3.     d^  d^  c^  a^  is  even. 

2.  a^b^  C2  ^3  is  odd.  4.     d^  b^  Ci  a<^  is  odd. 

Is  each  of  the  following  products  even  or  odd  ? 

5.  a^  ^3  Cr,  4  ^1.  7.     br^  d^  ei  c^  a^. 

6.  ^5  ^4  b^  e^  ^2-  2.     4  ^1  (Tg  rt-g  <^4- 

435.  A  square  array  of  11^  elements  arranged  in  n 
columns  and  n  rows  is  called  a  Determinant  of  the  ;/th 
Order,  71  denoting  any  whole  number.  The  general 
symbol  for  a  determinant  of  the  /^th  order  is  A. 

436.  The  Expansion  of  A  is  the  algebraic  sum  of 
all  the  different  products  that  can  be  formed  by 
taking  as  factors  one,  and  only  one,  element  from 
each  column  and  each  row  of  A,  and  giving  to  each 
product  the  coefficient  + 1  or  —  i  according  as  the 
product  is  even  or  odd. 

Cor.  If  each  element  of  a  cohnnn  {or  row)  of  ^  is 
zero,  A  =  0. 

Example  i.  Show  that  the  special  rules  for  expanding  de- 
terminants of  the  second  order  and  those  of  the  third  agree  with 
the  general  definition  given  above. 

Example  2.  Prove  that  one  half  of  the  products  in  the 
expansion  of  A  are  even  and  have  +  i  as  a  coefficient,  and  the 
other  half  are  odd  and  have  —  i  as  a  coefficient. 


DETERMINANTS.  331 

437.  The  expansiojt  of  ^  luis  \n  terms. 

For  after  the  first  element  of  a  product  has  been 
taken  from  the  first  column  in  any  one  of  the  7t  pos- 
sible ways,  the  second  element  can  be  taken  from 
the  second  column  in  any  one  of  «  —  I  ways,  and  so 
on.  Hence  a  product  of  7i  elements  can  be  obtained 
from  A  in  \n  different  ways  (§  339)  ;  therefore  there 
are  \n  terms  in  the  expansion  of  A. 

Cor.     Any  given  element  occurs  in  \n  —  \  terms. 

Thus,  if  A  is  of  the  6th  order,  its  expansion  contains  [6,  or 
720,  terms;   and  each  element  occurs  in  [5,  or  120,  terms. 

438.  If  in  A  the  cohimns  are  changed  into  corre- 
sponding rows,  the  resulting  determinant  A'  =  A. 

For  if  we  expand  A  by  taking  the  first  element  in 
each  product  from  the  first  column,  the  second  ele- 
ment from  the  second  column,  and  so  on;  and  then 
expand  A'  by  taking  the  first  element  in  each  product 
from  the  first  row,  the  second  element  from  the  sec- 
ond row,  and  so  on;  we  shall  obtain  the  same  terms 
in  the  expansion  of  A'  as  of  A  ;   hence  A'  =  A. 

Thus, 


^1 

h 

^1 

=     «1 

^2 

az 

«2 

^2 

^2 

^1 

b2 

bz 

«3 

^3 

Cs 

^1 

C2 

Cz 

Cor.  Whatever  theorem  is  proved  in  regard  to  the 
columns  of  a  determinant  is  true  also  in  regard  to  the 
rowSy  and  vice  versa. 


332 


ALGEBRA. 


439.  If  in  A  any  two  adjacent  rows  {or  columns) 
are  interchanged^  the  resulting  determinant  A'  =  —  A, 
^r  A  =  —  A'. 

For  interchanging  any  two  adjacent  rows  of  A  will 
simply  interchange  two  adjacent  subscript  numbers  in 
each  term  of  its  development;  hence  the  coefficient 
of  each  product  in  its  development  will  be  changed 
from  +  I  to  —  I,  or  from  —  i  to  +  i  ;   .*.  A'  =  —  A. 

Thus, 


^1 

h 

^1 

=  - 

^2 

h 

^2 

^2 

h 

^2 

H 

h 

^1 

^3 

K 

^3 

H 

h 

^3 

440.  If  in  A  any  two  rows  (or  columns)  are  inter- 
changed,  the  resulting  determinant  A'  =  —  A. 

Let  there  be  m  rows  between  the  two  rows  in 
question. 

The  lower  row  can  be  brought  just  below  the  upper 
one  by  m  interchanges  of  adjacent  rows;  the  upper 
row  can  then  be  made  to  take  the  place  first  held  by 
the  lower  one  by  m  +  i  interchanges  of  adjacent 
rows :  hence  in  all  there  will  be  2  7n  +  i  interchanges 
of  adjacent  rows,  or  the  determinant  will  change  the 
sign  of  its  coefficient  2  7n  +  i  times. 

.-.  A'  =  (_i)-«  +  'A  =  — A. 


Thus, 


^1 

^1 

=:  — 

^3 

h 

Cz 

^o 

c^ 

«2 

h 

C-1 

h 

^3 

'''^ 

h 

^1 

DETERMINANTS.  333 

441.  If  two  rows  {or  columns)  of  A  are  identical 
A  =  0. 

For  if  the  two  identical  rows  of  A  are  interchanged, 
the  resulting  deterniinant  is  —  A.  But  since  the  inter- 
changed rows  are  identical,  the  two  determinants  are 
identical ;  that  is,  A  =  —  A,    .-.  2  A  =  0,  or  A  =  0. 

442.  A  is  sometimes  written  in  the  form 

A=  a{  a{'  al" ^^ 

ai  ai'  ai" d^^ 


in  which  the  subscript  denotes  the  row  as  before,  and 
the  superscript  the  column  to  which  any  element 
belongs.  Thus  the  element  dh  belongs  to  the  //th  row 
and  the  ^th  column. 

In  any  determinant  the  element  at  the  upper  left- 
hand  corner  is  called  the  Leading  Element,  and  the 
place  which  it  occupies  the  Leading  Position. 

Any  element  d^^  may  be  brought  into  the  leading 
position  by  iji  —  i)  interchanges  of  adjacent  rows, 
and  {k  —  i)  interchanges  of  adjacent  columns.  The 
resulting  determinant  A'  will  equal  (—  t)''"^*~^A,  or 
(_  i)^'  +  *A  (§  439),  which  equals  A  or  —  A  according 
as  //  +  /^  is  even  or  odd. 

443.  If  each  eleine7it  of  a  column  {or  rotv)  of  A  be 
multiplied  by  any  number  m,  tJie  resulting  determinant 
A'  =  m  A. 


334 


ALGEBRA. 


For  each  term  in  the  expansion  of  A'  will  evidently 
be  m  times  the  corresponding  term  in  the  expansion 
of  A. 

444.  If  the  elements  of  any  column  {or  row)  of  A 
have  a  common  ratio  m  to  the  corresponding  elements 
of  any  other  column  {or  row),  then  A  =  0. 

For  then  A  —  m  times  a  determinant  that  vanishes 
(§§  443,441). 


Thus, 


a^     ci\     b-^ 


do     be, 


=  0. 


445.  If  each  element  of  any  columjt  {or  row)  of  A  is 
the  sum  of  two  or  more  members,  A  can  be  zuritten  as 
the  sum  of  two  or  moj'e  determinants.     That  is, 

I  ^1   +   ^1   +  ^1  ^2  ^3  I  =  kl  ^2  '^S  I  +  I  ^1  4  «?3  I  +  kl  4  ^3  I  . 

For  each  term  in  the  expansion  of  A  will  evidently  be 
equal  to  the  sum  of  the  corresponding  terms  in  the 
expansion  of  the  two  or  more  determinants. 

The  converse  of  this  principle  is  sometimes  useful  in  adding 
determinants.     Thus, 


3     4     5 

+ 

3 

-7     5 

+ 

3     3     5 

m 

4     7     7 

4 

-8     7 

4     I     7 

2       2       6 

2 

-3     6 

2      I      6 

7  +  3  5 
8+1  7 
3+1     6 


=  0. 


446.  If  each  element  of  any  column  (or  rozv)  of  A, 
or  each  element  multiplied  by  m,  be  added  to  or  sub- 
tracted from  the  corresponding  element  of  any  other 


DETERMINANTS. 


335 


coluni7i  {or  row)  the  resulting  determinant  A'  =  A. 
For  A'  c*an  always  be  written  as  the  sum  of  A  and 
another  determinant  which  vanishes. 


Thus, 


«1 

^ 

'l 

:^ 

«2 

h 

^2 

^3 

h 

^8 

=    ^1     ^1  ±  w  ^1     c^ 


bo  ±  ?''/  ^. 


(I) 


Vox  Ui      <^2  ±  '''  ^2      <^8  I  =  I  ^1  ^^2  <^8  I  =t  W  I  ^1  C\  Cq  I  §  443. 

This  principle  is  very  useful  in  simplifying  and   expanding 
determinants. 


Thus, 


Again, 


I 

I 

a    b  +  c 
b     c  -\-a 
c    a  +  b 

— 

I     «    a 
I     b     a 
I     r     « 

+  b  +  c 
+  b  +  c 

5     3     3 
5     I     4 
5     5     2 

5    9    3 
5    9    4 
5    9    2 

=  0. 

=  0.     §  444- 


Here  we  increase  each  element  of  the  second  column  by  twice 
the  corresponding  element  of  the  third  column. 


447.  If  A  is  a  rational  integral  function  of  2^  and  of 
b,  sue! I  that  if  b  be  substituted  for  a,  A  =  0,  then  a  —  b 
is  a  factor  of  A. 

For  the  expansion  of  A  may  be  written  in  the  form 

A  =  ^0  +  ^1  ^  +  y^2  ^^  +  ^3 «"  +  •  •  •       ( I ) 

in  which  A^,  Au  A^,  •  •  •  are  independent  of  a.     Since, 
when  b  is  substituted  for  ^,  A  =  0,  we  have 

0  =  Ao-^^i^  +  Ab'  +  A,b'-\----        (2) 


336 


ALGEBRA. 


Subtracting  (2)  from  (i),  we  obtain 
A  =  A  (^-^)  +  A,  {a'-b'')  +  ^3  {ci^  -  h^)  +  ••• 
in  which  form  A  is  evidently  divisible  by   a  —  b. 


I     a 

«2 

I     b 

^2 

I      c 

^2 

Example.     Factor  A  — 


A  =  0  when  b  =  a,  when  c  =  a^  and  when  ^  =  <^,  §  441  ;  hence 
A  contains  the  factors  b  —  a,  c  —  a,  and  c  —  <^.  The  product 
of  these  three  factors  is  of  the  same  degree  as  A,  and  has  the 
same  coefficients  as  the  expansion  of  A ;  hence  we  have 

L^{b-a)(c-a){c-  b). 

448.  Minors  and  Co-factors.  If  we  cancel  both  the 
row  and  the  column  of  A  to  which  any  element,  as 
a^^f\  belongs,  the  uticancelled  elements  of  A  form  a 
determinant  which  is  called  the  Minor  of  the  element 
<?i'^'\  and  is  denoted  by  A^(^). 

Thus  in  the  determinant  (i),  a{     a{'    a{" 

Aa,'  =  I  <'  <"  I,  Aa,'  =  I  a^"  <'  I,         <     a^"     a^"     (r) 
A.3'  =  I  a{'  a^"  I,  A«,''  =  I  a{  a^"  |,  •  •  •    a^     a^'     a^" 

Example.     If  A  is  of  the  4th  order,  write  out  A«/',  A^^', 

A«3'",  Aa^iv,  A^/,  A^Jv,  A^^'". 

(_  1)^  +  ^  A^^)  is  called  the  Co-factor  of  the  element 
aSf^,  and  is  denoted  by  A^^\ 

Cor.  Hence  the  co-factor  of  the  element  aiJ"^  equals 
-\-  or  —  its  minor  according  as  \i  -\-  \i  is  even  or  odd. 

Thus,  A^'  =  -Aa,",  /^e"  =  ^^o'^  ^3  =  -^/'3.  Q  =  ^c,. 


DETERMINANTS.  337 

449.  Terms  containing  an  Element.  The  minor 
A^/  is  a  determinant  of  the  {n  —  i)th  order;  hence 
its  expansion  contains  \n  —  i  terms.  Hence  Ui  Aa{' 
includes  all  the  \n  —  i  products  in  A  that  involve  ai, 
and  no  others.  Prefixing  a/  to  any  product  in  A,,/ 
does  not  change  the  number  of  its  inversions.  Hence 
the  coefficient  of  any  term  in  A^/  is  the  same  as  that 
of  the  corresponding  term  in  A ;  therefore  ai  A^j' 
includes  the  \n  —  i  terms  in  A  that  involve  rt/,  and  no 
others. 

If  the  element  ^y/*  is  transferred  to  the  leading 
positiott,  the  resulting  determinant  is  (—  i)'*"'"*A 
(§  442).     By  what  precedes,  it  follows  that 

cth^  ArtW  =  the  sum  of  the  terms  o{\—  i)  ^  +  *A  which 
contain  ^y/"'; 

.-.  (-  ly^^af  t^a^^  =  the  sum  of  the  terms  of  A 
which  contain  4f'. 

That  is,  the  sum  of  all  the  terms  of  ^  which  contain 
the  element  of  the  hth  row  and  the  kth  column  is  the 
product  of  that  element  by  its  co-factor, 

450.  Expression  of  A  in  Co-factors.  From  the  de- 
finition of  its  expansion,  A  equals  the  sum  of  all  the 
terms  that  contain  an  element  from  any  given  column 
or  row.     Hence  if  A  =  \ax  b^  c.^  ^4 1,  we  have 


^  =  b,B,  +  b,  B,  +  b,  B,  +  b,  B 


or  A  =  -/^i  A^^  +  ^2  ^b.,  -  b^  A^3  +  b^  A^4. 

22 


33  S  ALGEBRA. 

Again,      ^  =  a^A^-\-  b^B^  +  c,  C^  +  4  A, 


or 


By  this  principle  any  determinant  can  be  expressed 
as  the  sum  of  determinants  of  the  next  lower  order. 
Thus, 


6    2    3 

=  6 

4     5 

-  2 

2     3    +3 

2    4     5 

2    4 

2    4 

324 

2     3  !  =  26. 

4     5I- 


If  all  the  elements  except  one  in  any  column  or 
row  of  A  are  zero,  A  is  equal  to  that  element  into  its 
co-factor.     Thus, 


2    7 

6 

5 

E 

I     I 

I 

3 

I     5 

3 

4 

4    7 

5 

6 

254-1 

100       o 
142       I 

431-6 


=:  — 

5     4 

-  I 

4     2 

I 

3     I 

-6 

=  -45. 


The  second  determinant  is  obtained  from  the  first  by  sub- 
tracting the  elements  of  the  first  column  from  the  elemepts  of  the 
second  and  from  those  of  the  third,  and  three  times  the  elements 
of  the  first  from  those  of  the  fourth. 


In  §  428,  if  we  multiply  equation  (i)by  the  co-factor 
of  a^  in  the  determinant  of  system  (<2),  multiply  (2) 
by  the  co-factor  of  <^2»  (3)  by  the  co-factor  of  a^,  and 
add  the  resulting  equations,  we  obtain  (.4).  Hence, 
by  §  160,  (4)  gives  the  true  value  o( x. 

Example.  In  §  429  explain  how  equation  (5)  is  obtained 
from  equations  (i),  (2),  (3),  and  (4). 


DETERMINANTS. 


339 


EXERCISE  69. 


Show  that 


2. 


8. 


1203 
2312 
0321 
3123 

I  3  I  o 
3231 
0251 
2223 


=  -37' 


-85. 


3213 

4224 

2316 

10  4  5  8 

1224 
I  4  4  I 
I  I  2  2 
4  8  II  ^3 


=  0. 


=  15- 


2-159 
3  3  3  II 
2312 

5   7  3  7 


■yi 


6. 


2  7—2 

4  I    I 
03—1 

6  4 


=  14. 


3 

4 

2  -8 


X2    Z2 

+  J2 

Xi       Zi 

-ys  -^1   ^1 

rz 

Xs       Z-s 

X3    Z3 

X^      Z2 

Xi 

Jl 

^1 

X2 

/2 

2^2 

•^3 

J3 

^3 

I    a   a" 

(^ 

I    /^    I?' 

b' 

I    ^    r^ 

c^ 

I    d  d' 

d^ 

a 

b  -\-  c    d^ 

b 

a-\-  c    IP- 

c 

a^b     ^ 

=  {b-a){c-d){c-b){(i-d){d-b)(d-c). 


=  {a-\-b-{-c)  (a-b)  (a-c)  (b  —  c). 


First  add  each  element  of  the  first  column  to   the   corre- 
sponding clement  of  the  second. 


340 


ALGEBRA. 


Solve  the  equation 


lO. 


2X 

5 

X 

3 

Zoc 

2 

2 

3 

2,x 

==  0. 


II. 


X 


3     3 
3 

.%•"      2        I 


3     -^ 

,2 


=  0. 


Solve  the  system 


12. 


■X  i-y  +  2  +  w 
x—y  \-  z  ■\-  w 
x+y  —  z 
x+y  +  z 


—  ze/=  2. J 


i3« 


15- 


-  + =  ^. 


i6.  4:v  +  yjF  +  3^—  2  7<!/=  9, 
2  j:  —  7  —  40  +  3  7£/  =  13, 
^x-\-2y~'jz  — 4.w=  2, 
5^  —  3y  +     z  +  S'^=  13-. 

For  other  examples  see  Exercise  12,  page  84. 


X 

+  2J 

— 

z 

=    OA 

y 

+    z 

— 

2  W 

=  o,( 

z 

—    X 

+ 

2  W 

=  2S,( 

2  X 

+  ^w 

+ 

f 

=  5>J 

2 
X 

-'-  + 

y 

4 

z 

:    2.9, J 

5_ 

X 

y 

10 

•4  = 

Z    { 

2  +  1^ -_ 
y     ^ 

8 

X 

= 

14.9.) 

451.  Eliminants.  When  the  number  of  equations 
in  a  system  is  greater  by  one  than  the  number  of  un- 
knowns, the  system  is  impossible,  that  is,  its  eqiiatious 
are  inconsistent,  unless  the  relation  between  the  co- 
efficients is  such  as  to  make  one  equation  of  the 
system  depend  upon  the  others. 


DETERMINANTS. 


341 


Let  us  consider  the  following  system  of  three  linear 
equations  involving  only  two  unknowns. 
aix  +  b^y  +  nix  =  0,     (i)  \ 
a^x  -{-  d.^y  +  ;//2  =  0,     (2)  >  {a) 

a^x  +  b^y  +  m.  =  0.     (3)) 

Multiplying  equations  (i),  (2),  (3),  respectively  by 
the  co-factors  of  ;«i,  m2,  m^,  in  the  determinant 
1^1  b-i  m^y  and  adding  the  resulting  equations,  we 
obtain 


"1 

J^3 


a^x  -\r  biy  +  mi 


ai     bi 
^9     bo 


a^x  -\-  b^y  +  W2 

a^x  ■\- b^y -\r  nn         a^     b^     0 

.*.  I  a^  b^  Ph  I  =  0.  (4) 

If  the  equations  in  {a)  are  consistent,  the  relation 
in  (4)  must  hold  true ;  conversely,  if  (4)  is  satisfied, 
the  equations  in  {a)  are  consistent. 

Note  that  |  a^  b^  m^  \  is  the  determinant  of  the  coeffi- 
cients and  the  known  terms  in  system  {a). 

Example.     Test  the  consistency  of  the  equations 

X  ^-    y  +  2z  =  g,' 
x+    y~    2  =  0, 
7.x-    y+     2  =  3, 
X  —  2y  +  22  =  l._ 

Here  the  determinant  of  the  coefficients  and  the  known  terms 
is  found  to  equal  zero ;  hence  the  equations  are  consistent. 

Equation  (4)  is  called  the  Eliminant  or  Resultant 
of  system  (a),  because  it  is  the  result  obtained  by 
elimmating  the  unknowns  from  its  equations. 


i<i) 


342  ALGEBRA. 

452.  A  homogeneous  linear  equation  is  one  in  which 
every  term  is  of  the  first  degree;  thus  a^x  -{-  b^y  + 
C\Z  —  ^  is  a  homogeneous  linear  equation.  Let  us 
consider  the  following  system  of  three  homogeneous 
linear  equations  involving  three  unknowns, 
a^x  ^  b^y  ■\-  c^z  —  0,  (i)  \ 
a^x  +  b2y  +  CzZ  —  O,         (2)  V  (a) 

asx  +  bsy  +  CsZ  =  0.         (3)) 

If  the  equations  are  independent,  the  only  solution 
of  system  (^)  is  x  =  0,  y  =  0,  ;s  —  0  ;  for  the  numer- 
ator in  the  value  of  each  unknown  is  zero  (§  443 
Cor.). 

If  however  laibiC^]  —  0,  the  value  of  each  un- 
known assumes  the  indeterminate  form  0  -f-  0,  and 
the  system  is  indeterminate  (§§  174,  176).  Hence 
I  ^1  ^2  ^3 1  —  0  expresses  the  condition  that  one  equa- 
tion of  system  (a)  shall  depend  on  the  others.  When 
this  condition  is  fulfilled,  the  system  is  essentially  one 
of  two  equations  with  three  unknowns.  In  this  case 
the  ratios  of  the  unknowns  may  be  found  by  trans- 
posing the  last  term  in  (i)  and  (2)  to  the  second 
member,  and  then  solving  these  two  equations  for  x 
and  y.     We  thus  obtain 

X  =  z\—eib2\-^\aib2\,y  =  z\ai  —  e2\-^\aib2\ 
.' .  X  :  y  :  z  ::  \  —  Ci  b2\  :  \  ai  —  ^2]  :  \  ai  b2\. 

Example.     Is  the  system         x  —    y  —  22  =  0,  (i)  ^ 

x-2y+     2'  =  0,  (2)  >■  (a) 

2x-3y-    2  =  0,  (3)) 
indeterminate?     If  so,  find  the  ratios  of  x,y,  z. 


DETERMINANTS. 


343 


The  determinant  of  the  system  equals  zero  ;  hence  the  system 
is  indeterminate.     Solving  (i)  and  (2)  for  x  and  _y,  we  obtain 


453.    Multiplication  of  Determinants.      Let 


A  = 


o      o 
o      o 


I 
o 

X2 


(0 


then 


^1 

b,       0 

■—  I 

— 

aA 

0      Xi        jFi 

0     X2       y-i 

1 

^1       «2 

X 

^1  71 

^1 

Jh 

^2 

yi 

—  I 

0 

x^ 

yi 

x^ 

y% 

(2) 


Increasing  each  element  of  the  first  column  in  (i) 
by  a^  times  the  corresponding  element  of  the  third 
column  plus  a^  times  the  corresponding  element  of 
the  fourth  column ;  in  like  manner  increasing  each 
element  of  the  second  column  by  b\  times  that  of  the 
third //;/^  b^_  times  that  of  the  fourth  ;  we  then  have 
A  = 


0 

0 

—  I 

0 

0 

0 

0 

—  I 

cixx^  +  ciiyx 

bxx  ^  h^yx 

Xx 

yi 

^,^2+    ^2^2 

\  x^  +  hyi 

^2 

y^ 

ax  Xx  +  a^yx 

hx  xx  +  b^yx 

^1^2  +  ^2j^'2 

bx  X2  +  ^2  y^ 

(3) 


Equating  the  values  of  A  in  (2)  and  (3),  we  have 


xxyx 


X 


^1  ^2  = 

bxb,\ 


(txXx  +  a^yx      bxXx-^b^yx 
ax  x-i  +  a^y^      bx  x^  +  b^  y^ 


(4) 


344 


ALGEBRA. 


Each  column  of  elements  in  the  product  may  be 
obtained  as  follows :  Multiply  the  elements  of  the 
first  row,  Xi,  yi,  by  ai,  a2  respectively,  and  take  their 
sum  ;  then  multiply  the  elemefits  of  the  seeottd  row,  Xg^ 
y.,,  by  ai,  a^  respectively,  and  take  their  sum  ;  these  sums 
are  the  elements  of  the  first  column  in  the  product. 
The  elements  of  the  second  column  in  the  product  are 
found  in  a  similar  manner,  by  using  the  elements  of  the 
second  row  of  the  multiplier  instead  of  those  of  the  first. 


By  letting     A 


a\     bx 

«2 


b, 

^3       ^3 


^1     —I 
^■2 


I 

O 

o 

o    - 

-  I 

o 

o 

o 

—  I 

Xi 

yi 

Zl 

^2 

i'2 

22 

•^8 

yz 

Zz 

we  may  in  like  manner  prove  that 


^1 

yi 

Zl 

X 

^2 

J'2 

Z-2 

^8 

J'3 

Zz 

^1        <^2 

bi     b^ 

Ci        C2 


ai  Xi  -\-  a^y^  +  a^  Zi  bi  x^  +  ^2  Ji  +  ^3  Zi  c^  x^  +  <^2  J^i  +  ^3  -^i 
ax  X.2  +a2y2  +  ^3  2^2  ^i  -"^2  +  <^2  J2  +  bs  z.^  c^  x^  -f  c^y^.  +  c^  z^ 
^1  ^3  +  ^2^3+^3  Zz    bi  xz  +  biyz  +  b^  03    ^1  ^3  +  c^y^  +  ^3  2^3 


(5) 


The  same  method  of  proof  will  apply  to  a  deter- 
minant of  any  order.  Hence  the  rule  given  above 
will  enable  us  to  obtain  one  form  of  the  product  of 
two  determinants. 


DETERMINANTS. 


345 


Thus  we  have 

3     4 

X 

4    3 

= 

2    3 

5    4 

232 

X 

2     I     2!E| 

322 

I      2      3 

I     3 

I 

2 

2    4 

12+12    15  +  16  j  —  24    31 

8+9  10  +  12  I  17  22 
4  +  3  +  4  2  +  6  +  6  4  +  6  +  8 
6  +  2  +  4  3  +  4  +  6  6  +  4  +  8 
2  +  3  +  2     1+6  +  3     2  +  6  +  4 


=  0.) 


EXERCISE    70. 

Find  the  ratios  of  the  unknowns  in  the  system 
I.     — 4x+y  +  z  =  0,  i  2.     2  X  +  jy  —  2Z 

X—  2^  +  2  =  0.  )  y  +  4.Z  —  4W 

jr—  5J'+     2     +   2W 

3.  U aix  +  l>iy  +  Ciz  =  a2X  -\-  dzy  +  C2Z  =  a^x  +  b^y 
+  ^sZ  =  a^x  -{-  If^y  +  c^z ;  find  the  relation  that  must  exist 
between  the  coefficients. 

Put  each  trinomial  equal  to  —  « ;  then  the  required  relation 
will  be  1  rtj  <^2^8  ^1  =  0-  This  relation  expresses  the  condition 
that  the  resulting  system  shall  be  indeterminate. 

4.  Eliminate  x  from  px^  +  gx-\-r=0,  (i) 

aindax^ -{- dx^ -\- ex -{- d=0.  (2) 

From  (i),  px'^  +  gx  +  r=0. 


(a) 


From  (2), 


ax^  +  bx'^^-cx^d-  0. 


id) 


.  Multiply  (2)  by  ,r,     ax^  +  b x^  +  cx'^  +  dx        =0. 
Multiply  (i)  by  Jir,  px^ -\- q x"^  +  rx        =0. 

Multiply  (i)  by  x^  p  x^  ■}-  qx^  +  rx"^  =  0. 

These  five  equations  involve  the  four  unknowns  x^,  x^,  x'^f 
x;  hence,  by  §  451,  their  eliminant  is 


0 

0 

P 

Q 

r 

0 

a 

b 

c 

d 

a 

b 

c 

d 

0 

0 

P 

^ 

r 

0 

P 

i 

r 

0 

0 

=  0. 


(3) 


346 


ALGEBRA. 


Equations  (i)  and  (2)  being  consistent,  equation  (3)  is  their 
eliminant ;  or,  to  test  the  consistency  of  (i)  and  (2),  ascertain 
whether  or  not  (3)  is  satisfied. 

EHminate  x  from  the  consistent  equations 
5.    ax^ -\-  bx  +  c  =  0,  \       6.    ax^-\-bx!^-{-cx-\-d  =  ^, 


x^  \  qx  ■\-  r  ^^.) 


p  x^  ■\-  qx^ 


+  rx-{-  J-  =0.  ) 


7.    ax'^ -^  bx^ -\- cx^ -\- dx -\-  e  ^=^, 


=  0.f 


p  x^  -^  qx  ■\-  r 

Test  the  consistency  of  the  equations 

8.  2x^  —  2>x'-\-  6a:  =  0, 

x'^  —  2  X  —   3   =0. 

9.  dx^  —  3  x'^y  —  xy^  —  1 2  j^  =  0,  ^ 

4^^  —Zxy-^  3 j/2  —  0.  3 

Divide  the  first  equation  by  x^  and  the  second  by  x^^  then 
regard  the  different  powers  of  j  -f  at  as  the  unknowns. 


10.    Find  the  product  of 


II. 


Find  the  product  of 


I 

2 
I 

7 
2 


15 

7 


1  2 

2  I 

and 


Note  that 


12.    Find  the  product  of 


and 


GRAPHIC    SOLUTIONS. 


347 


CHAPTER    XXIII. 

GRAPHIC    SOLUTION    OF    EQUATIONS    AND 
SYSTEMS. 


454.  Let  XX'  and  Y  Y'  be  two  fixed  straight  lines 
at  right  angles  to  each  other  at  O.  Let  OX  and 
6^  F  be  positive  directions;  then  OX'  and  O  Y'  will 
be  negative  directions.  The  lines  XX'  and  Y  Y' 
divide  their  plane  into  four  equal  parts  called 
Quadrants,  which  are 
numbered  as  follows: 
The  First  Quadrant 
is  X  0  Yy  the  Second 
YOX',  the  T/iird 
X'  O  F.  and  the  ^'' 
Fourt/i  Y'OX. 

The  lines  XX'  and 
YY',  are  called  Axes 
of  Reference,  and  their 
intersection  O  the 
Origin. 

P>om  P,  any  point  in  the  plane  of  the  axes,  draw 
PJf  parallel  to  Y  Y' ;  then  the  position  of  P  will  be 
determined  when  we  know  both  the  leiigths  and  the 
directions  of  the  lines  6^  J/ and  MP, 


Fig.  I. 


348  ALGEBRA. 

The  line  O  M  \s  called  the  Abscissa  of  the  point  P; 
and  MP  is  called  the  Ordinate  of  P.  The  abscissa 
and  ordinate  together  are  called  the  Co-ordinates  of  P. 

Thus,  O  A  and  A  P'  are  the  co-ordinates  oi  F'  \  the  abscissa, 
O  A^  is  negative,  and  the  ordinate,  A  /",  is  positive.  O  C,  the 
abscissa  of  F'",  is  positive,  and  C F"\  its  ordinate,  is  negative. 

An  abscissa  is  usually  denoted  by  the  letter  x^  and 
an  ordinate  by  j.  The  axis  XX'  is  called  the  Axis 
of  Abscissas,  or  Axis  of  x;  and  Y  V,  the  Axis  of  Ordi- 
nates,  or  Axis   of  y. 

The  point  whose  co-ordinates  are  x  and  j/  is 
written  (x,j/). 

Thus  (2,  —3)  denotes  the  point  of  which  the  abscissa  is  2, 
and  the  ordinate  —  3.  We  use  a  system  of  co-ordinates  analo- 
gous to  that  explained  above  whenever  we  locate  a  city  by- 
giving  its  latitude  and  longitude.  In  this  case,  the  equator  is 
one  axis,  and  the  assumed,  meridian  the  other. 

Example.     Construct  the  point  (-2,  3)  ;  (-3,  —  4). 

In  Fig.  I,  lay  off  (9  /^  =  —2,  and  on  A  F'  parallel  to  V  V  lay 
off  /?  /"  =  +3  ;  then  is  P'  the  point  (—2,  3). 

To  construct  (—  3,  —4),  lay  off  (9^  =  —3,  and  on  B F"  par- 
allel to  YV  lay  off  B  F"  =  -4;  then  is  F"  the  point  (-3,-4). 

EXERCISE   71. 

1.  Construct  the  point  (2,  3);  (4,7);  (3,-5);  (-2,-3); 
(4,-2);   (-5,-3);   (-2,  5). 

2.  In  which  quadrant  is  the  point  (^',  j^^),  when  .x  and  _y 
are  both  positive  ?  both  negative  ?  x  negative  and  y  positive  ? 
X  positive  and  y  negative  ? 


GRAPHIC    SOLUTIONS.  349 

3.  In  which  quadrant  may  (x,  j)  be,  when  x  is  positive? 
X  negative  ?  y  positive  ?  y  negative  ? 

4.  In  what  Hne  is  (x^  o)  ?  (o,  y)  ?  Where  is  (o,  o)  ?  (4,  o)  ? 
(-3,0)?   (0,2)?   (0,-5)? 

455.  To  solve  an  indeterminate  equation  graphically 
is  to  draw  the  line  or  lines  which  include  all  the 
points,  and  only  those,  whose  co-ordinates  satisfy  the 
equation.  The  line  or  lines  is  called  the  Graph  of  the 
equation. 

456.  To  draw  the  graph  of  an  indeterminate  equa- 
tion, we  obtain  a  number  of  its  solutions,  then  con- 
struct a  sufficient  number  of  the  corresponding  points 
to  determine  the  form  of  the  graph,  and  through 
these  points  trace  a  continuous  curve. 

Example  i.     Solve  y^x^-x-^y  graphically. 
If  we  put  ;r  =  —3,  —2,  ...  ,  we  obtain 

when    x-  —3,   —2,   —  i,       o,  i,       i,       2,   3,   4,... 

j)/=      6,       o,   -4,  -6,   -6^,  -6,  -4,   o,   6,... 

Drawing  the  axes  XX'  and  Y  V  in  Fig.  2,  and  assuming  O  i 
as  the  linear  unit,  we  construct  the  points  (—3,  6),  (-2,  o), 
(-1,-4),...  Tracing  a  continuous  curve  through  these 
points,  we  obtain  the  curve  A  a  B  2.^  the  graph  of  the  given 
equation. 

As  X  increases  from  4,  j  or  :r2  —  ;r  —  6  continues  positive 
and  increases;  hence  there  is  an  infinite  branch  of  the  locus  in 
the  first  quadrant.  As  x  decreases  from  —  3,  j  continues  posi- 
tive and  increases;  hence  there  is  an  infinite  branch  in  the 
second  quadrant. 


350 


ALGEBRA. 


Example  2.     Solve  j  =  x^  —  2x  graphically. 
When  x=  —2,   —  i,   —0.8,   o,       0.8,       i,   2,... 

y=    -4,        I,         I.I,    o,    -I.I,    -I,    4,  ..  . 

Locating  these  points  as  in  Fig.  3,  and  tracing  a  continuous 
curve  through  them,  we  obtain  the  curve  MNP  Q  as  the  graph 
oiy  =  x^  ~  2  X. 

Here  evidently  one  infinite  branch  is  in  the  first,  and  the 
other  in  the  third  quadrant. 


Fig.  2. 


Whenever  there  is  any  doubt  about  the  form  of  a  graph  be- 
tween any  two  determined  points,  intermediate  points  should 
be  located. 


Example  3.     Construct  the  graph  of/ 


3  :r2  +  4. 


When  :r=      -f, 

y  =   —6.1 
The  5:raph  is  given  in  Fig.  4 


I,   -1,   o,     \,     I,    2,    3,  .  .  . 
o,    3.1,   4,    3.4,    2,  o,   4,  .  .  . 


GRAPHIC    SOLUTIONS. 


351 


Example  4.     Solve   y  =  x^-\-x^  —  'ix'^—x-\-2  graphically. 
When  x=  -|,   -2,     - 1,     -i,  _^,   o,      \,    i,      |, 

j=    9.2,       o,  —1.6,       o,    1.7,   2,   09,  o,    2.2. 
The  graph  is  given  in  V\g.  5. 

Of  the  infinite  number  of  real  solutions  of  an  inde- 
terminate equation  each  is  represented  geometrically 


by  the  co-ordinates  of  some  point  in  its  graph  ;  hence 
the  graph  of  an  indeterminate  equation  represents 
geometrically  all  its  infinite  number  of  real  solutions. 


EXERCIS: 

Solve  graphically  the  equation 


y  ^=  x""—  2  X 


y 


7^-+  10. 


y  —  X^—:^X^  + 

y  —  x^  -\-  4X  + 


352  ALGEBRA. 

5.  Construct  two  points  of  the  graph  of  the  linear  equa- 
tion y  =  2  X  -\-  ;^,  and  prove  that  the  unlimited  straight  line 
through  them  is  the  graph  of  this  equation. 

Similarly  the  graph  of  any  linear  equation  can  be  con- 
structed ;  hence  t/ie  graph  of  any  linear  equatiofi  in  x  atid  y 
is  a  straight  line. 

6.  Construct  the  graph  oi  y  —  ^iX—  2)  of2_y  =  —  4^+  i; 
of  37  +  5 -^'+2=0;  of:v  =  3;  ofa:  =  — 5;  of  7  =  2  ;  of 

y  =  -^• 

The  equation/  —  b  may  be  written  in  the  form  y  —  ox  -\-  b^ 
in  which,  for  any  value  q>{  x^y  =  b;  hence  the  graph  of_7  =  ^  is  a 
line  parallel  to  the  axis  of  x. 

7.  Show  that  the  graph  of  the  equation  j*;  ^  +  _>'^  —  5-  is  a 
circle  whose  centre  is  at  the  origin,  and  whose  radius  is  5. 

Evidently  the  graph  of  any  equation  of  the  form  x^  +y^  =  r^ 
is  a  circle  whose  centre  is  at  the  origin  and  whose  radius  is  r. 

8.  Construct  the  graph  of  ^^  +  y  =  9  ;  oi  x^  -{■  y"^  =  16  \ 
of  2^^  +  2jv^  =  8. 

9.  Construct  the  graph  of  4  jr^  +  9  ^  =  36. 


Herej  =  ±|  Vg  -  x\ 

Evidently  —3  is  the  least  real  value  of  x  that  will  render  y 
real ;  hence  no  part  of  the  graph  can  lie  to  the  left  of  the  line 
:r  =  — 3.  For  like  reason  no  part  of  the  graph  can  lie  to  the 
right  of  jr  =  3. 

When;r=:— 3,    —2.5,    —2,       — i,       o,         i,  2,        3, 

j  =  ±o,   ±1.1,   ±1.5,   ±1-9'   ±2,   ±1.9,   ±1.5,   ±0. 

The  graph  is  the  ellipse  ANBS  (Fig.  9,  page  356),  the 
semi-axes  being  3  and  2. 


GRAPHIC   SOLUTIONS. 


353 


lo.    Construct  the  graph  oiy^—4x  —  x^ 


/' 


Herey=-xy  .      ^, 

The  graph  is  the  curve  H P  P' A  (Fig.  8,  page  355). 

It  is  evident  that  when  x  —  v>,y  approaches  -x\  hence,  in 
either  quadrant,  the  infinite  branch  approaches  indefinitely  near 
to,  but  cannot  reach  the  line/  —  -x,  or  D  B. 

1 1.  Show  that  the  graphs  oi y  =  2x  -\-  i  and  _y  =  2  a:  +  3 
intercept  equal  segments  on  lines  parallel  to  Y  V\  and  are 
therefore  parallel.  Show  also  that  y  =  x  and  y  =  x  +  c  are 
parallel,  and  that  each  makes  an  angle  of  45°  with  the  axis 
of  X. 

12.  Construct  the  graph  ofy=  4  jc;  of  4^— 97^=36; 
of  (i  +  x^)  y  =  X. 


a'x+c'.      (2))  ^  ^ 


457.    Graphic  Solution  of  Systems  of  Equations. 

Example  i.     Solve  the  system  / 

/  = 

Let  the  graph  of  (i)  be  the  straight  line  MB,  and  that  of 
(2)    the    line    R  P; 

then  the  single  solu-  ^M 

tion  of  system  {ii)  is  ^ 

the  co-ordinates  of 
the  common  point  P. 
This  illustrates  the 
general  truth  that  a 
system  of  linear  equa- 
tions has  one  and 
only  one  solution. 

By  measuring  the 
co-ordinates  O  A  and 

A  P,  the  numerical  solution  of  the  system  may  be  obtained 
approximately. 

23 


Fig.  6. 


354 


ALGEBRA. 


(a) 


li  a  =  a'  and  <:  =  ^',  the  graphs  of  (i)  and  (2)  will  evidently  co- 
incide ;  that  is,  the  graphs  of  equivalent  equations  coincide. 

\i  a  —  a\  and  c  =  not  c',  the  graphs  of  (i)  and  (2)  will  be 
parallel ;  for  they  will  intercept  equal  segments  of  the  value 
c  —  c\  on  all  lines  parallel  to  Y  Y' j  that  is,  the  graphs  of  in- 
consistent linear  equations  are  parallel. 

As  the  lines  PR  and  P M  approach  parallelism,  their  inter- 
section P  recedes  to  an  infinite  distance  from  the  origin.  When 
they  become  parallel,  their  intersection  is  lost  at  infinity,  which 
illustrates  the  infinite  solution  of  an  impossible  linear  system 
(Example  of  §  176). 

Example  2.     Solve  the  system  x^  -\-  y'^  =  25,         (i) 

y-x  =  c.  (2)  / 

If  O  A  =5,  the  graph  of  (i)  is  the  circle  P' P'"  P,  and,  if 

c  —  ^.,  the  graph  of 
(2)  is  the  straight 
line  M  Nj  hence 
the  two  solutions 
of  system  {a)  are 
the  co-ordinates  of 
the  two  points,  P 
and  P'. 

By  measurement 
we  find  the  two  so- 
lutions to  be 

;r  =  — V-,j/  =  -2/. 
For  ^  =  5  |/2  or 
"  _  5  |/2,   the   graph 

Fig.  7.  of  (2)  is  the  tan- 

gent    N'    M'     or 
N''  M'\  and  the  two  solutions  of  the  system  are  equal. 

For  c  <  5  \/2  and  >  —  5  \/2,  the  graph  of  (2)  lies  between 
N' M'  and  N"  M" ^  and  the  two  solutions  of  the  system  are 
real  and  unequal. 


/ 

^ 

^ 

— ^^       /^ 

A 

/!\ 

N^ 

^1 

/            ^ 

\ 

v' 

X 

A 

X- 

\ 

k 

■^^ 

'o 

y^'" 

X 

GRAPHIC    SOLUTIONS. 


355 


=  4-^-^,  (I) 

=  ax  i-  c.     (2) 


[(«) 


For  c  >  5  I  '2  or  <      542,  the  graph  of  (2)  does  not  cut  the 
circle,  and  both  solutions  of  the  system  are  imaginary. 

Example  3.     Solve  the  system      y' 

y 

The  graph  of  (i)  is  the 
curve'  H  P  O  F'  A,  of 
which  the  infinite  branches 
approach  indefinitely  near 
to  the  graph  of  ^  =  —  x, 
Qx  D  B  (Example  10  of 
Exercise  72). 

\ia—  I,  and  c=  0,  the 
graph  of  (2)  is  the  line 
M N^  and  the  three  solu- 
tions of  system  {a)  are 
the  co-ordinates  of  the 
points  P,  (9,  and  P' . 

If  ^  =  —  I,  and  ^  =  0, 
the  graph  of  (2)  is  DB, 

and  system  {a)  is  defective;  only  one  solution  is  finite,  the  other 
two  being  infinite. 

\i  a  —  —  \  and  r  =  2,  the  graph  of  (2)  is  A' A',  and  two 
solutions  of  {a)  are  finite,  the  third  being  infinite. 

If  c  were  increased  from  2,  the  two  finite  solutions  would 
approach  equality,  become  equal,  and  then  become  imaginary. 

If  in  equation  (i)  we  put  v  =  —  x.  we  obtain 

-  ;i-8  =  4 r  -  .r8,  or  o .r«  +  o.r^  -  4  :r  =  0, 
which  illustrates  §  407,  since  the  abscissas  of  both  P  and  P' 
become  infinite,  when  the  line  M N  is  revolved  clockwise  about 
O  to  the  position  of  D  B. 

Example  4.     Solve  the  system    ^x^  ^  gy^  =  36,  (i) 

X^  i-     j2  —  ^2^ 

The  graph  of  (i)  is  the  ellipse  A  NBS,  in  which  O  A  =  2, 
and  ON—  2. 


Fig.  8. 


356 


ALGEBRA. 


If  r  =  |,  the  graph  of  (2)  will  be  the  circle  P  F  P" ,  and  all 
four  solutions  of  the  system  will  be  real  and  unequal. 

If  r  =  3,  the  circle  will  bj  tangent  to  the  ellipse  at  A  and  B; 
hence  two  solutions  of  the  system  will  be  ;ir  =  3,  j  =  0,  and  the 
other  two  jr  =  —  3,  j/  =  0. 

If  r  =^  2,  the  circle  will  be  tangent  to  the  ellipse  at  A^and  6^. 

If  r  <  2  or  >  3,  the  two  graphs  will  have  no  common  points, 
and  all  the  solutions  of  the  system  will  be  imaginary. 


P^^_ 

_r_^ 

0           i 

pHn^^ 

^f^'" 

^—-'^y^ 

^^ 

Y 

Fig.  9. 

li  r  =  ^,  by  clearing  (2)  of  fractions,  and  then  subtracting  it 
from  (i)  we  obtain  5^/^  =  11,  of  which  the  graph  is  the  parallel 
lines  P  P'  and  P'"  F' .  These  lines  cut  either  the  ellipse  or  the 
circle  in  all  points  common  to  both  of  these  curves,  and  only 
in  these  points.  This  illustrates  the  equivalency  of  system  {a) 
to  the  system      4X^  +  gy^  =  36  >  x^  +  /^  =  V''   I 

458.  A  system  of  equations  involving  x  a7id  y,  one 
of  the  nth  degree  and  the  other  of  the  ist,  has  n  and 
onfy  n  solntions. 


GRAPHIC   SOLUTIONS.  35/ 

For  substituting  the  value  ofjj/,  as  obtained  from 
the  Hnear  equation,  in  the  equation  of  the  ;/th  degree, 
we  obtain  in  general  an  equation  of  the  ;^th  degree  in 
X.  This  equation  gives  ;/  values  for  x,  each  of  which 
gives  one  value  for  j/  in  the  linear  equation. 

459.  A  system  of  equations  involving  x  and  y,  one 
of  the  mth  degree  and  the  other  of  the  nth,  has  in  gen- 
eral mn  solutions. 

The  general  proof  of  this  theorem  is  too  long  and  difficult  to 
be  given  here.  The  theorem  is  very  evident,  however,  when 
one  of  the  equations  can  be  resolved  into  equivalent  linear 
equations- 

For  example,  the  system 

(x-2y-  I)  (x+y-2)  (^  +  3/)  =  0,  j        ^^^ 
is  equivalent  to  the  three  systems, 

X*  —  dy*  =  axy  \     x^  —  by^  =  axy\     x^  —  by^  =  axy} 
X—  2y  =  I         )'     X  -\-   y  =  2        )        :ir+37=0        > 

each  of  which  by  §  458  has  four  solutions;  hence  the  given 
system  (a)  has  4  x  3,  or  12,  solutions. 

460.  By  its  ordinates,  the  graph  of  j^==  ^  (^)  rep- 
resents graphically  the  continuous  series  of  values  of 
F  {x)  corresponding  to  a  continuous  series  of  values 
of  .r;  hence  the  graph  oiy  =  F(^x^  is  often  called 
the  graph  of  F  (:r). 

The  graph  of /^(;ir)  clearly  illustrates  the  following 
properties  of  F  (x}  and  of  the  equation  F  (x)  =  0. 


358  ALGEBRA. 

(i.)    The  abscissa  of  any  point  in  which  the  graph 

of  F  {x)  cuts  or  touches  the    axis  of  x  is 

one  of  the  unequal  or  equal   real   roots   of 

the  equation  F  (x)  =  0. 

Hence  the  real  roots  of  F  {x)  =  0  may  be  obtained 

approximately   by    measuring   the    abscissas    of  the 

points  in  which  the    graph  of  F  (x)   cuts  the   axis 

of  ;ir. 

At  a  point  of  tangency  the  graph  is  properly  said 
to  meet  the  axis  of  ;ir  in  two  coincident poirits. 

Thus  from  the  graph  in  Fig.  2  we  learn  that  one  root  of  the 
equation  :r'^  —  ;f  —  6  =  0  is  —  2,  and  the  other  is  3. 

In  Fig.  3,  the  graph  crosses  the  axis  of  x  between  ;r  =  —  2 
and  X  —  ~  \\  hence  one  root  of  x^  —  7.x  :=^  lies  between  -  2 
and  —  I ;  a  second  root  is  zero,  and  the  third  lies  between  i 
and  2. 

In  Fig  4,  the  graph  cuts  the  axis  of  ;ir  at  ;jr  =  —  1,  and 
touches  it  at  ;r=:  2;  hence  one  root  of  ^ir^  —  3  jr^  +  4  =  0  is  —  i, 
and  the  other  two  are  2  each. 

(ii.)  The  grc  ph  of  F  {x)  renders  evident  the 
theorem  of  §  397. 

For  it  is  clear  that  the  portion  of  a  continuous 
curve  between  any  two  points  must  cut  the  axis 
of  X  an  odd  number  of  times  when  these  points  are 
on  opposite  sides  of  that  axis ;  and  an  even  number 
of  times,  or  not  at  all,  when  the  points  are  on  the 
same  side  of  that  axis. 

Thus,  in  Fig.  3,  the  graph  cuts  XX'  an  odd  number  of  times 
between  M  and  /V,  or  M  and  Q,  and  an  even  number  of  times 
between  M  and  P. 


GRAPHIC   SOLUTIONS.  359 

In  Fig.  5,  the  graph  cuts  XX'  an  odd  number  of  times  be- 
tween M  and  N,  or  N  and  /4,  and  an  even  number  of  times 
between  M  and  R,  R  and  A,  ox  M  and  A. 

(ill.)  The  graph  of  F  {x)  also  illustrates  the  fact 
that  equal  roots  form  the  connecting  link 
between  real  and  imaginary  roots,  and  that 
imaginary  roots  occur  in  pairs. 

Thus,  by  slightly  diminishing  the  absolute  term  4  of  the 
function  ;r8  —  3:^2 -f  4,  its  graph  in  Fig.  4  would  be  moved 
downward,  and  would  then  cut  the  axis  of  x  in  three  points  ;  by 
slightly  increasing  the  term  4,  the  graph  would  bs  moved  up- 
ward, and  would  then  cut  the  axis  of  x  in  but  one  point.  This 
illustrates  the  fact  that  the  two  equal  real  roots  of  the  equation 

^^  -  3  :f  2  +  4  =  0 
would   become  unequal  real  roots  or  imaginary,  according  as 
the  absolute  term  4  were  diminished  or  increased. 

By  increasing  the  absolute  term  of  F{x^  (Example  4,  §  456) 
by  2,  all  the  roots  oi  F{x)  —  0  would  become  imaginary. 

(iv.)    The  graph  of  F  (a-)   illustrates   §§  398  and 

399. 
For  when  F  (,r)  is  of  an  odd  degree,  one  infinite 
branch  of  its  graph  will  be  in  the  first  quadrant,  and 
the  other  in  the  third;   hence  the  graph  will  cut  the 
axis  of  X  in  at  least  one  point. 

When  F  {x^  is  of  an  even  degree,  one  infinite 
branch  of  its  graph  will  be  in  the  first  quadrant,  and 
the  other  in  the  second.  Now  if  pn  is  negative,  the 
graph  will  cut  the  axis  of  jj/  below  the  origin;  hence 
it  will  cut  the  axis  oi x  in  at  least  two  points,  one  to 
-^the  right  and  the  other  to  the  left  of  the  origin. 


36o 


ALGEBRA. 


EXERCISE   73. 

Construct  the  graph  of  F  (x)  and  locate  the  real  roots  of 
F  ix)  —  ^  in  each  of  the  following  examples  : 

\.  F  {x)^o(^  ^  X    —2.        ^.  F  {x)^x^  —  ix^—\x-^\\. 

2.  F{x)^x^-2x-^.        5.  >^(jt:)  =  ^3-4^-6^+8. 

3.  F{x)^x'^  —  ^x^^\o.     6.  ^(.t)  =  :^*— 4^^  — 3X+23. 

7.  i^  (^)  =  ^^  -h  2  :r^  —  3  ^-^  —  4  -^  +  4- 

8.  F(x)  =  a;^  +  4^^—  14  a;^  —  17^  —  6. 


461.  Geometric  Representation  of  Imaginary  and 
Complex  Numbers.  A  line  whose  value  includes  both 
its  length  and  direction  is  called  a  Directed  Line,  or 

a  Vector.  We  pro- 
ceed to  show  that 
any  algebraic  num- 
ber, real,  imagi- 
nary, or  complex, 
can  be  represented 
by  some  vector. 

Let  vector  O  A 
represent  +  i  ;  then 
OA'  =  -i.  But 
(-M)x(-i)  =  -i; 
hence  —  i  as  a  fac- 
tor reverses  the  yq.c- 
ior  0  A,  or  turns  it  through  180°.  Therefore  +  y'-T, 
or—  \/^^(that  is,  one  of  the  two  equal  factors  of—  i) 


15 

i 

\ 

,-! 

+  1 

0 

1 

"3 

B' 

Fig.  10. 


GRAPHIC    SOLUTIONS.  36 1 

will  revolve  the  vector  O  A  through  90°.  Suppose 
V— I  to  revolve  the  vector  OA  counter-clockwise; 
then  — V—  I  vvill  revolve  it  clockwise :  hence  O  B 
=  +  V^i  and  O  B'  =  -  V^T. 

For  brevity,  the  symbol  V—  i  is  generally  denoted 
by  t.  Hence  t  as  a  factor  revolves  a  vector  through 
90°  counter-clockwise,  and  therefore  —  t  revolves  a 
vector  90°  clockwise.     Hence  we  have 

(i)  ii=   {+1)   ll:=-I, 

Hi—  (-i-  i)  //•  i  —  (—  i)  /,  or  —  /, 
iiii—{-\- 1)  ii-  it  =  (—  i)  //  =  -f  i. 
(ii)       ai  =  ia)  that  is,  (-h  \)ai—  (-h  i)  ia,         (i) 

For  multiplying  the  unit  by  a  and  then  revolving 
it  through  90°  gives  the  same  result  as  first  revolving 
the  unit  through  90°  and  then  multiplying  it  by  a. 

ai    bi=  ab  '  ii,  a  ii  •  b,  or  //'  •  a  b,  (2) 

For  multiplying  the  vector  ai  hy  b  and  then  re- 
volving it  through  90°  gives  the  same  result  as 
revolving  the  vector  ^^  through  90°  twice  in  succes- 
sion, as  multiplying  the  vector  aii  (or  —  a)  by  b,  or 
as  multiplying  the  vector  ii  (or  —  i)  hy  ab. 

That  is,  the  commutative  and  associative  laws  of 
multiplication  hold  true  for  imaginary  factors. 

Next  let  us  consider  the  complex  number  a  -\-  b  i, 
where  a  and  b  are  both  real. 


362 


ALGEBRA. 


Let  a  —  \  ^2,  and  b 


B 

^-'''     T    ^~--^ 

P>                      !                     XP 

/ 

\                       1                        / 

/ 

\                    1                    / 

\ 

/ 

\                 1                 / 

/ 

\               '               / 

/ 

\.            !           / 

\ 

/ 

/ 
a'I 

\       1       / 
\  1  / 

\ 

M'                 /0\                   M 

/       1      \ 
/          1        \ 

\ 

\ 

/             1             \ 

/ 

\ 

/                1               \. 

/ 

\ 

/                       [                     \ 

/ 

\ 

/ 

P"X                I                /?'" 

^--~,     i   .-"" 

\  \/2.  Lay  o^  O  M=\  /^2, 
and  on  MP  drawn 
perpendicular  to 
OA  lay  off  J/P  = 
I  ^2  ;  then  0  M  — 
+  i  V2,     M  P  = 


\  V2  •  ^"• 


V'2 


B' 
Fig.  II. 


+  \^2'i  ^  0 M 
+  MP=  OP.  (i) 
The  vector   OP 
equals  the  sum  of 
the    vectors    O  M 
and  MP;  for  trans- 
ference from  O  to 
M     followed      by 
transference  from  M  to  P  gives  the  same  result  as 
transference  from  0  directly  to  P.     For  like  reason 

OP"'r=^        \^2-\^2i,  (2) 

OP^    =-\^2-V\^2i,  (3) 

OP"  ^-h^/2-\  ^2i.         (4) 

In  like  manner  any  complex  number  may  be 
represented  by  some  vector. 

In  Fig.  10,  the  vectors  O  B  and  OB'  represent  the 
two  square  roots  of  —  i  ;  for  either  multiplied  by 
itself  gives  —  I. 

In  Fig.  II,  by  geometry  we  know  that  OP,  O P\ 
OP",  OP"'  are  each  a  unit  in  length.    Hence  J  ^/2 


GRAPHIC    SOLUTIONS.  363 

(i  +  /)  as  a  factor  revolves  the  unit  O  A  through  45°; 

whence  [^  ^2  (i  +  0?  ==  ^^' =  -  I-  (S) 
As  a  factor  \^^2  (i  —  i)  revolves  OA  through  —  45°  i 

whence  l\  ^2{i  -  i)]"^  =  0  A' = -i.  (6) 
As  a  factor  --  ^ /^2  (i  —  i)  revolves   OA  through 

135°; 

whence  [- ^  ^2  (i  —  t)f  =  0  A' =  -  i.  (7) 
As  a  factor  —  ^  ^2  {i  +  i)  revolves  OA  through 

-135°; 

whence      [- \  ,^2  {\  -\- i)f  =  O  A' = -i.      (8) 

Hence  OP,  OP',  OP",  OP'"  represent  the  four 
fourth  roots  of  —  i. 

In  like  manner  we  could  represent  geometrically 
the  six  sixth  roots  of  —  i  ;    and  so  on. 


SCIENCE.  47 

Physics  for  Uniuersity  Students. 

By  Professor  Henry  S.  Carhart,  University  of  Michigan. 

Parti.     Mechanics,  Sound,  and  Light.     With  154  Illustrations.     i2mo, 

cloth,  330  pages.     Price,  ^1.50. 

Part  II.     Heat,  Electricity,  and  Magnetism.     With  224  Illustrations. 

i2mo,  cloth,  446  pages.     Price,  $1.30. 

THESE  volumes,  the  outgrowth  of  long  experience  in  teach- 
ing, offer  a  full  course  in  University  Physics.  In  preparing 
the  work,  the  author  has  kept  constantly  in  view  the  actual  needs 
of  the  class-room.  The  result  is  a  fresh,  practical  text-book,  and 
not  a  cyclopaedia  of  physics. 

Particular  attention  has  been  given  to  the  arrangement  of 
topics,  so  as  to  secure  a  natural  and  logical  sequence.  In  many 
demonstrations  the  method  of  the  Calculus  is  used  without  its 
formal  symbols ;  and,  in  general,  mathematics  is  called  into  ser- 
vice, not  for  its  own  sake,  but  wholly  for  the  purpose  of  establish- 
ing the  relations  of  physical  quantities. 

It  is  believed  that  the  work  will  be  helpful  to  teachers  who 
adopt  the  prevailing  method  of  a  combination  of  lectures  and 
text-book  instruction.  As  it  is  intended  to  supplement,  not  super- 
sede, the  teacher,  it  leaves  ample  scope  for  the  personal  equation 
in  instruction. 

Professor  W.  LeConte  Stevens,  Rensselaer  Polytechnic  Institute,  Troy,  N.  Y.  : 
After  an  examination  of  Carhart's  University  Physics,  I  have  unhesitat- 
ingly decided  to  use  it  with  my  next  class.  The  book  is  admirably 
arranged,  clearly  expressed,  and  bears  the  unmistakable  mark  of  the 
work  of  a  successful  teacher. 

Professor  Florian  Cajori,  Colorado  College :  The  strong  features  of  his  Uni- 
versity Physics  appear  to  me  to  be  conciseness  and  accuracy  of  statement, 
the  emphasis  laid  on  the  more  important  topics  by  the  exclusion  of  minor 
details,  the  embodiment  of  recent  researches  whenever  possible. 

Professor  A.  A.  Atkinson,  OAio  University,  Athens,  O. :  I  am  very  much 
pleased  with  the  book.  The  important  principles  of  physics  and  the 
essentials  of  energy  are  so  well  set  forth  for  the  student  for  which  the  book 
is  designed,  that  it  at  once  commends  itself  to  the  teacher. 

Professor  Sarah  F.  Whiting,  Wellesley  College,  Mass. :  I  am  using  it  with 
one  of  my  classes,  and  find  that  it  is  well  adapted  to  supplement  lectures 
and  to  put  the  student  in  possession  of  salient  points. 


48  SCIENCE. 


The  Elements  of  Physics. 

By  Professor  HENRY  S.  Carhart,  University  of  Michigan,  and  H.  N. 
Chute,  Ann  Arbor  High  School.    i2mo,  cloth,  392  pages.    Price,  ^1.20. 

THIS  is  the  freshest,  clearest,  and  most  practical  manual  on 
the  subject.     Facts  have  been  presented  before  theories. 

The  experiments  are  simple,  requiring  inexpensive  apparatus, 
and  are  such  as  will  be  easily  understood  and  remembered. 

Every  experiment,  definition,  and  statement  is  the  result  of 
practical  experience  in  teaching  classes  of  various  grades. 

The  illustrations  are  numerous,  and  for  the  most  part  new, 
many  having  been  photographed  from  the  actual  apparatus  set 
up  for  the  purpose. 

Simple  problems  have  been  freely  introduced,  in  the  belief 
that  in  this  way  a  pupil  best  grasps  the  application  of  a  principle. 

The  basis  of  the  whole  book  is  the  introductory  statement 
that  physics  is  the  science  of  matter  and  energy,  and  that  noth- 
ing can  be  learned  of  the  physical  world  save  by  observation  and 
experience,  or  by  mathematical  deductions  from  data  so  obtained. 
The  authors  believe  that  immature  students  cannot  profitably  be 
set  to  rediscover  the  laws  of  Nature  at  the  beginning  of  their 
study  of  physics,  but  that  they  must  first  have  a  clearly  defined  idea 
of  what  they  are  doing,  an  outfit  of  principles  and  data  to  guide 
them,  and  a  good  degree  of  skill  in  conducting  an  investigation. 

WiJliam  H.  Runyon,  Armour  Institute,  Chicago  :  Carhart  and  Chute's  text- 
book in  Physics  has  been  used  in  the  Scientific  Academy  of  Armour 
Institute  during  the  past  year,  and  will  be  retained  next  year.  It  has 
been  found  concise  and  scientific.  We  believe  it  to  be  the  best  book  on 
the  market  for  elementary  work  in  the  class-room. 

W.  C.  Peckham,  Adelphi  Academy,  Brooklyn,  N.  Y. :  Carhart  and  Chute's 
Physics  on  the  whole  impresses  me  as  the  best  book  for  a  beginner  to 
use  in  getting  his  first  view  of  the  general  principles  of  the  whole  subject. 

Professor  A.  L.  Kimball,  Amherst  College :  As  a  text-book  to  be  used  in 
high  school  classes,  I  do  not  know  of  any  that  is  superior  to  it. 

Professor  C.  T.  Brackett,  Princeton  University:  I  have  examined  this 
work  with  care  and  with  pleasure,  for  it  presents  the  fundamental  prin- 
ciples of  physics  with  exactness  and  with  clearness. 

Professor  George  F.  Barker,  University  of  Pennsylvania  :  The  book  is  an 
excellent  one ;  the  best  of  its  grade  in  the  market. 


SCIENCE.  49 


Electrical  Measurements, 

By  Professor  Henry  S.  Carhart  and  Asst.  Professor  G.  W.  Patter- 
son, University  of  Michigan.     i2mo,  cloth,  344  pages.     Price,  $2.00. 

IN  this  book  are  presented  a  graded  series  of  experiments  for 
the  use  of  classes  in  electrical  measurements.  Quantitative 
experiments  only  have  been  introduced,  and  these  have  been 
selected  with  the  object  of  illustrating  general  methods  rather 
than  applications  to  specific  departments  of  technical  work. 

The  several  chapters  have  been  introduced  in  what  the  authors 
believe  to  be  the  order  of  their  difificulty  involved.  Explana- 
tions or  demonstrations  of  the  principles  involved  have  been 
given,  as  well  as  descriptions  of  the  methods  employed. 

The  Electrical  Engineer,  New  York :  We  can  recommend  this  book  very 

highly  to  all  teachers  in  elementary  laboratory  work. 
The  Electrical  Journal,  Chicago :  This  is  a  very  well-arranged  text-book 

and  an  excellent  laboratoi7  guide. 

Exercises  in  Physical  Measurement 

By  Louis  W.  Austin,  Ph.D.,  and  Charles  B.  Thwing,  Ph.D., 
University  of  Wisconsin.     i2mo,  cloth,  198  pages.     Price,  ^1.50. 

THIS  book  puts  in  compact  and  convenient  form  such  direc- 
tions for  work  and  such  data  as  are  required  by  a  student 
in  his  first  year  in  the  physical  laboratory. 

The  exercises  in  Part  I.  are  essentially  those  included  in  the 
Practicum  of  the  best  German  universities.  They  are  exclu- 
sively quantitative,  and  the  apparatus  required  is  inexpensive. 

Part  II.  contains  such  suggestions  regarding  computations  and 
important  physical  manipulations  as  will  make  unnecessary  the 
purchase  of  a  second  laboratory  manual. 

Part  III.  contains  in  tabular  form  such  data  as  will  be  needed 
by  the  student  in  making  computations  and  verifying  results. 

Professor  Sarah  F.  Whiting,  Wellesley  College :  It  comprises  very  nearly 
the  list  of  exercises  which  I  have  found  practical  in  a  first-year  college 
course  in  Physics.  I  note  that  while  the  directions  are  brief,  skill  is 
shown  in  seizing  the  very  points  which  need  to  be  emphasized.  The 
Introduction  with  Part  II.  gives  a  very  clear  presentation  of  the  essential 
things  in  Measurements,  and  of  the  treatment  of  errors. 


60  SCIENCE. 

Principles  of  Chemical  Philosophy. 

By  JosiAH  Parsons  Cooke,  late  Professor  of  Chemistry,  Harvard 
University.     Revised  Edition.    8vo,  cloth,  634  pages.     Price,  ^3.50. 

THE  object  of  this  book  is  to  present  the  philosophy  of  chem- 
istry in  such  a  form  that  it  can  be  made  with  profit  the 
subject  of  college  recitations.  Part  I.  of  the  book  contains  a 
statement  of  the  general  laws  and  theories  of  chemistry,  together 
with  so  much  of  the  principles  of  molecular  physics  as  are  con- 
stantly applied  to  chemical  investigations.  Part  II.  presents  the 
scheme  of  the  chemical  elements,  and  is  to  be  studied  in  con- 
nection with  experimental  lectures  or  laboratory  work. 

Elements  of  Chemical  Physics. 

By  JosiAH  Parsons  Cooke.    8vo,  cloth,  751  pages.    Price,  $4.50. 

THIS  volume  furnishes  a  full  development  of  the  principles 
of  chemical  phenomena.  It  has  been  prepared  on  a 
strictly  inductive  method  and  any  student  with  an  elementary 
knowledge  of  mathematics  will  be  able  easily  to  follow  the  course 
of  reasoning.  Each  chapter  is  followed  by  a  large  number  of 
problems. 

Chemical  Tables. 


By  Stephen  P.  Sharples.    izmo,  cloth,  199  pages.    Price,  ^2.00. 

Logarithmic  and  Other  Mathematical  Tables. 

By  William  J.  Hussey,  Professor  of  Astronomy  in  the  Leland  Stan- 
ford Junior  University,  California.    8vo,  cloth,  148  pages.     Price,  ^i.oo. 

IN  compiling  this  book  the  needs  of  computers  and  of  students 
have  been  kept  in  view.  Auxiliary  tables  of  proportional 
parts  accompany  the  logarithmic  portions  of  the  book,  and  all 
needed  helps  are  given  for  facilitating  interpolation. 

Various  mechanical  devices  make  this  work  specially  easy  to 
consult ;  and  the  large,  clear,  open  page  enables  one  readily  to 
find  the  numbers  sought. 


SCIENCE.  51 


Anatomy,  Physiology,  and  Hygiene. 

"  A  Manual  for  the  Use  of  Colleges,  Schools,  and  General  Readers.    By 
Jerome  Walker,  M.D.    i2mo,  cloth,  427  pages.    Price,  ^1.20. 

THIS  book  was  prepared  with  special  reference  to  the  require- 
ments of  high  and  normal  schools,  academies,  and  colleges, 
and  is  believed  to  be  a  fair  exponent  of  the  present  condition  of 
the  science.  Throughout  its  pages  lessons  of  moderation  are 
taught  in  connection  with  the  use  of  each  part  of  the  body. 
The  subjects  of  food,  and  of  the  relations  of  the  skin  to  the 
various  parts  of  the  body  and  to  health,  are  more  thoroughly 
treated  than  is  ordinarily  the  case.  All  the  important  facts  are 
so  fully  explained,  illustrated,  and  logically  connected,  that  they 
can  be  easily  understood  and  remembered.  Dry  statements  are 
avoided,  and  the  mind  is  not  overloaded  with  a  mass  of  technical 
material  of  little  value  to  the  ordinary  student. 

The  size  of  type  and  the  color  of  paper  have  been  adopted  in 
accordance  with  the  advice  of  Dr.  C.  R.  Agnew,  the  well-known 
oculist.  Other  eminent  specialists  have  carefully  reviewed  the 
chapters  on  the  Nervous  System,  Sight,  Hearing,  the  Voice,  and 
Emergencies,  so  that  it  may  justly  be  claimed  that  these  impor- 
tant subjects  are  more  adequately  treated  than  in  any  other  school 
Physiology. 

The  treatment  of  the  subject  of  alcohol  and  narcotics  is  in 
conformity  with  the  views  of  the  leading  physicians  and  physiol- 
ogists of  to-day. 

The  Nation,  New  York :  Dr.  Jerome  Walker's  Anatomy,  Physiology,  and 
Hygiene  appears  an  almost  faultless  treatise  for  colleges,  schools,  and 
general  readers.  Careful  study  has  not  revealed  a  serious  blemish ;  its 
tone  is  good,  its  style  is  pleasant,  and  its  statements  are  unimpeachable. 
We  cordially  commend  it  as  a  trustworthy  book  to  all  seeking  information 
about  the  body,  and  how  to  preserve  its  integrity. 

Journal  of  the  American  Medical  Association:  For  the  purposes  for 
which  it  is  written,  it  is  the  most  interesting  and  fairest  exponent  of 
present  physiological  and  hygienic  knowledge  that  has  ever  appeared. 
It  should  be  used  in  every  school,  and  should  be  a  member  of  every 
family,  —  more  especially  of  those  in  which  there  are  young  people.  It  is 
a  pleasure  to  read  and  review  such  an  excellent  book. 


SCIENCE. 


The  Elements  of  Chemistry. 


By  Professor  Paul  C.  Freer,  University  of  Michigan.     i2mo,  cloth, 
294  pages.     Price,  ^1,00. 

IN  the  preparation  of  this  book  an  attempt  has  been  made  to 
give  prominence  to  what  is  essential  in  the  science  of 
Chemistry,  and  to  make  the  pupil  famiHar  with  the  general 
aspect  of  chemical  changes,  rather  than  to  state  as  many  facts 
as  possible.  To  this  end  only  a  few  of  the  most  important 
elements  and  compounds  have  been  introduced ;  and  the  work, 
both  in  the  text  and  in  the  laboratory  appendix,  has  been  made 
quantitative. 

Chemical  equations  have  been  sparingly  used,  because  they 
are  apt  to  give  the  pupils  false  notions  of  the  processes  they 
attempt  to  record.  Considerable  space  has  been  given  tb  physi- 
cal chemistry,  and  a  constant  effort  has  been  made  to  present 
chemistry  as  an  exact  science. 

The  apparatus  required  to  perform  successfully  the  experi- 
ments suggested  will  not  be  found  expensive,  the  most  costly 
being  such  as  will  form  part  of  the  permanent  equipment  of  a 
laboratory,  and  if  properly  handled  will  not  need  to  be  replaced 
during  a  long  term  of  years. 

Professor  Charles  Baskerville,  University  of  North  Carolina:  It  is  the 
most  excellent  book  of  the  character  which  has  ever  come  to  my  notice. 
It  is  clear,  scientific,  and  thoroughly  up  to  date. 

Professor  E.  E.  Slosson,  University  of  Wyoming-:  Freer's  Elementary 
Chemistry  gives  the  most  completely  experimental  and  logical  proof  of 
the  fundamental  laws  of  chemistry  for  beginners  that  I  have  seen.  It 
teaches  chemistry  as  a  rational,  and  not  merely  as  a  descriptive,  science. 

Willard  R.  Pyle,  Media  Academy,  Media,  Pa. :  It  emphasizes  the  nature 
of  chemical  changes,  connects  facts,  and  leads  the  student  to  think.  In 
these  respects  it  is  superior  to  any  elementary  chemistry  I  know  of. 

Lewis  B.  Avery,  High  School,  Redlands,  Cat. :  The  book  has  proved  far 
better  than  I  had  dared  to  hope  for  in  adopting  it.  The  rational  mode 
of  treatment  and  the  remarkable  perspicuity  of  the  work  encourage 
sound  scientific  thought  and  genuine  interest  among  pupils  such  as  I 
have  not,  in  ten  years'  experience  with  the  best  of  the  books  now  most 
in  use,  been  able  to  obtain. 


SCIENCE.  53 


Descn'ptiue  Inorganic  General  Chemistry. 

A  text-book  for  colleges,  by  Professor  Paul  C.  Freer,  University  of 
Michigan.     Revised  Edition.    8vo,  cloth,  559  pages.     Price,  $3.00. 

IT  aims  to  give  a  systematic  course  of  Chemistry  by  stating 
certain  initial  principles,  and  connecting  logically  all  the 
resultant  phenomena.  In  this  way  the  science  of  Chemistry 
appears,  not  as  a  series  of  disconnected  facts,  but  as  a  harmo- 
nious and  consistent  whole. 

The  relationship  of  members  of  the  same  family  of  elements 
is  made  conspicuous,  and  resemblances  between  the  different 
families  are  pointed  out.  The  connection  between  reactions  is 
dwelt  upon,  and  where  possible  they  are  referred  to  certain  prin- 
ciples which  result  from  the  nature  of  the  component  elements. 

The  frequent  use  of  tables  and  of  comparative  summaries  les- 
sens the  work  of  memorizing  and  affords  facilities  for  rapid  refer- 
ence to  the  usual  constants,  such  as  specific  gravity,  melting  and 
boiling  points,  etc.  These  tables  clearly  show  the  relationship 
between  the  various  elements  and  compounds,  as  well  as  the  data 
which  are  necessary  to  emphasize  this  relationship.  They  also 
exhibit  the  structural  connection  between  existing  compounds. 

Some  descriptive  portions  of  the  work,  which  especially  refer 
to  technical  subjects,  have  been  revised  by  men  who  are  actively 
engaged  in  those  branches.  In  the  Laboratory  Appendix  will 
be  found  a  list  of  experiments,  with  descriptive  matter,  which 
materially  aid  in  the  comprehension  of  the  text. 

Professor  Walter  S.  Haines,  I^usA  Medical  College,  Chicago  :  The  work  is 
worthy  of  the  highest  praise.  The  typography  is  excellent,  the  arrange- 
ment of  the  subjects  admirable,  the  explanations  full  and  clear,  and  facts 
and  theories  are  brought  down  to  the  latest  date.  All  things  considered, 
I  regard  it  as  the  best  work  on  inorganic  chemistry  for  somewhat  advanced 
general  students  of  the  science  with  which  I  am  acquainted. 

Professor  J.  H.  Long,  Northwestern  University,  Evanston,  III.  :  I  have 
looked  it  over  very  carefully,  as  at  first  sight  I  was  much  pleased  with 
both  style  and  arrangement.  Subsequent  examination  confirms  the  first 
opinion  that  we  have  here  an  excellent  and  a  very  useful  text-book.  It 
is  a  book  which  students  can  read  with  profit,  as  it  is  clear,  systematic, 
and  modern. 


56  MA  THE  MA  TICS. 


An  Academic  Algebra. 


By  Professor  J.  M.  TAYLOR,  Colgate  University,  Hamilton,  N.Y.    i6mo, 
cloth,  348  pages.     Price,  ^i.oo. 

THIS  book  is  adapted  to  beginners  of  any  age  and  covers 
sufficient  ground  for  admission  to  any  American  college  or 
university.  In  it  the  fundamental  laws  of  number,  the  literal 
notation,  and  the  method  of  solving  and  using  the  simpler 
forms  of  equations,  are  made  familiar  before  the  idea  of  alge- 
braic number  is  introduced.  The  theory  of  equivalent  equa- 
tions and  systems  of  equations  is  fully  and  clearly  presented. 
Factoring  is  made  fundamental  in  the  study  and  solution  of 
equations.  Fractions,  ratios,  and  exponents  are  concisely  and 
scientifically  treated,  and  the  theory  of  limits  is  briefly  and 
clearly  presented. 

Professor  C.  H.  Judson,  Furman  University,  Greenville,  S.C.:  I  regard 
this  and  his  college  treatise  as  among  the  very  best  books  on  the  subject, 
and  shall  take  pleasure  in  commending  the  Academic  Algebra  to  the 
schools  of  this  State. 

Professor  E.  P.  Thompson,  Miami  University,  Oxford,  O. :  The  book  is 
compact,  well  printed,  presenting  just  the  subjects  needed  in  preparation 
for  college,  and  in  just  about  the  right  proportion,  and  simply  presented. 
I  like  the  treatment  of  the  theory  of  limit?,  and  think  the  student  should 
be  introduced  early  to  it.  I  am  more  pleased  with  the  book  the  more 
I  examine  it. 

E.  P.  Sisson,  Colgate  Academy,  Hatnilton,  N.  V. :  It  has  the  spirit  of  the 
best  modern  thought  on  mathematics.  The  book  is  conspicuously  meri- 
torious :  First,  In  the  clear  distinction  made  between  arithmetical  and 
algebraic  number,  which  lies  at  the  foundation  of  an  understanding  of 
Algebra.  Second,  The  introduction  at  the  very  first  of  the  equation  as  an 
instrument  of  mathematical  investigation.  By  this  instrument  many  of 
the  demonstrations  of  the  theorems  which  follow  have  a  conciseness  and 
clearness  which  could  not  otherwise  be  obtained.  Third,  Dr.  Taylor's 
presentation  of  the  doctrine  of  equivalency  is  clear  and  rigid.  Fourth, 
The  treatment  of  the  subject  of  factoring  is  concise,  comprehensive,  and 
logical.  ...     I  am  using  it  with  satisfaction  in  my  own  classes. 

Arthur  G.  Hall,  University  of  Michigan:  Its  clear,  concise,  and  logical 
presentation  renders  it  well  adapted  to  high  school  classes.  It  is  alto- 
gether the  best  text-book  for  secondary  schools  that  I  have  seen. 

The  Critic,  June,  rSgs  :  On  the  whole  the  book  is  the  best  elementary 
Algebra,  written  by  an  American,  that  has  come  to  our  notice. 


MA  THEM  A  TICS.  57 


A  College  Algebra. 


By   Professor  J.   M.   TAYLOR,  Colgate  University,   Hamilton,   N.Y. 
i6mo,  cloth,  326  pages.     Price,  ^1.50. 

A  VIGOROUS  and  scientific  method  characterizes  this  book. 
In  it  equations  and  systems  of  equations  are  treated  as 
such,  and  not  as  equalities  simply. 

A  strong  feature  is  the  clearness  and  conciseness  in  the  state- 
ment and  proof  of  general  principles,  which  are  always  followed 
by  illustrative  examples.  Only  a  few  examples  are  contained  in 
the  First  Part,  which  is  designed  for  reference  or  review.  The 
Second  Part  contains  numerous  and  well  selected  examples. 

Differentiation,  and  the  subjects  usually  treated  in  university 
algebras,  are  brought  within  such  limits  that  they  can  be  success- 
fully pursued  in  the  time  allowed  in  classical  courses. 

Each  chapter  is  as  nearly  as  possible  complete  in  itself,  so 
that  the  order  of  their  succession  can  be  varied  at  the  discre'.ion 
of  the  teachers. 

Professor  W.  P.  Durfee,  Hobart  College,  Geneva,  N,  Y. :  It  seems  to  me  a 
logical  and  modern  treatment  of  the  subject.  I  have  no  hesitation  in  pro- 
nouncing it,  in  my  judgment,  the  best  text-book  on  algebra  published  in 
this  country. 

Professor  George  C.  Edwards,  University  of  California:  It  certainly  is  a 
most  excellent  book,  and  is  to  be  commended  for  its  consistent  conciseness 
and  clearness,  together  with  the  excellent  quality  of  the  mechanical  work 
and  material  used. 

Professor  Thomas  E.  Boj'ce,  Middlehury  College,  Vt. :  I  have  examined 
with  considerable  care  and  interest  Taylor's  College  Algebra,  and  can  say 
that  I  am  much  pleased  with  it.  I  like  the  author's  concise  presentation 
of  the  subject,  and  the  compact  form  of  the  work. 

Professor  H.  M.  Perkins,  Ohio  Wesleyan  University:  I  think  it  is  an 
excellent  work,  both  as  to  the  selection  of  subjects,  and  the  clear  and 
concise  method  of  treatment. 

S.  J.  Brown,  Formerly  of  University  of  Wisconsin  :  I  am  free  to  say  that 
it  is  an  ideal  work  for  elementary  college  classes.  I  like  particularly  the 
introduction  into  pure  algebra,  elementary  problems  in  Calculus,  and  ana- 
lytical growth.  Of  course,  no  book  can  replace  the  clear-sighted  teacher ; 
for  him,  however,  it  is  full  of  suggestion. 


46  SCIENCE. 


Primary  Batteries. 


By  Professor  HENRY  S.  Carhart,  University  of  Michigan.    Sixty- 
seven  Illustrations.     i2mo,  cloth,  202  pages.     Price,  ^1.50. 

THIS  is  the  only  book  on  this  subject  in  English,  except  a 
translation.  It  is  a  thoroughly  scientific  and  systematic 
account  of  the  construction,  operation,  and  theory  of  all  the  best 
batteries.  An  entire  chapter  is  devoted  to  a  description  of  stand- 
ards of  electromotive  force  for  electrical  measurements.  An  ac- 
count of  battery  tests,  with  results  expressed  graphically,  occupies 
forty  pages  of  this  book. 

The  battery  as  a  device  for  the  transformation  of  energy  is 
kept  constantly  in  view  from  first  to  last ;  and  the  final  chapter 
on  Thermal  Relations  concludes  with  the  method  of  calculating 
electromotive  force  from  thermal  data. 

Professor  John  Trowbridge,  Harvard  University :  I  have  found  it  of  the 
greatest  use,  and  it  seems  to  me  to  supply  a  much  needed  want  in  the 
literature  of  the  subject. 

Professor  Eli  W.  Blake,  Brown  University  :  The  book  is  very  opportune, 
as  putting  on  record,  in  clear  and  concise  form,  what  is  well  worth  know- 
ing, but  not  always  easily  gotten. 

Professor  George  F.  Barker,  University  of  Pennsylvania :  I  have  read  it 
with  a  great  deal  of  interest,  and  congratulate  you  upon  the  admirable 
way  in  which  you  have  put  the  facts  concerning  this  subject.  The  latter 
portion  of  the  book  will  be  especially  valuable  for  students,  and  I  shall 
t)e  glad  to  avail  myself  of  it  for  that  purpose. 

Professor  John  E,  Davies,  University  of  Wisconsin  :  I  am  so  much  pleased 
with  it  that  I  have  asked  all  the  electrical  students  to  provide  themselves 
with  a  copy  of  it.  .  .  .  I  have  assured  them  that  if  it  is  small  in  size,  it 
is,  nevertheless,  very  solid,  and  they  will  do  well  to  study  and  work  over 
it  very  carefully.  .  .  .     I  find  it  invaluable. 

Albert  L.  Amer,  formerly  of  Iowa  State  University :  I  am  using  your  work 
on  Primary  Batteries,  and  find  it  the  best  guide  to  practical  results  in  my 
laboratory  work  of  anything  that  I  have  yet  secured.  It  is  a  book  we 
long  have  needed,  and  it  is  one  of  that  kind  which  is  not  exhausted  at 
a  first  reading. 

Professor  Alex.  Macfarlane,  University  of  Texas  :  Allow  me  to  congratu- 
late you  on  producing  a  work  which  contains  a  great  deal  of  information 
which  cannot  be  obtained  readily  and  compactly  elsewhere. 


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THE  UNIVERSITY  OF  CALIFORNIA  UBRARY 


